Zeta and q-Zeta Functions and Associated Series and Integrals

Zeta and q-Zeta Functions and Associated Series and Integrals

Zeta and q-Zeta Functions and Associated Series and Integrals

H. M. Srivastava Department of Mathematics University of Victoria Victoria Canada

Junesang Choi Department of Mathematics Dongguk University Gyeongju Republic of Korea

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Elsevier 32 Jamestown Road, London NW1 7BY 225 Wyman Street, Waltham, MA 02451, USA First edition 2012 c 2012 Elsevier Inc. All rights reserved. Copyright No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangement with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-385218-2 For information on all Elsevier publications visit our website at elsevierdirect.com This book has been manufactured using Print On Demand technology. Each copy is produced to order and is limited to black ink. The online version of this book will show color figures where appropriate.

Contents

Preface

xi

Acknowledgements

1

XV

Introduction and Preliminaries

1

1.1

1

Gamma and Beta Function The Gamma Function

1

Pochhammcr's Symbol and the Factorial Function

4

Legendre and Gauss n! and its Generalizations

Multiplication Formulas of Stirling's Formula for

The Beta

Function

7

The Incomplete Gamma Functions

10

The Incomplete Beta Functions

10

The Error Functions

11

The B o hr M o l l e rup Theorem

12

-

1.2

13

The Euler-Mascheroni Constant y A

et of Known Integral Representations for y

Further Integral Representations for y

From

1.3

an

Polygamma Functions Integral

Gauss's fonnulas for

24 25

lf!( �)

30

1/r(z)

31 33

The Polygamma Functions Spec ia l Value of

t<"l (z)

34

The Asymptotic Expansion for The Multiple Gamma

The

Functions

1/r(z)

36

38

Double Gamma Function r2

38

Integral Formulas involving the Double Gamma Function

45

The Multiple Gamma Functions

56

The

Evaluation of an Integral Involving log G(z)

The Triple Gamma Function r3

1.5

22 24

Digamma) Function Representations for 1/J(z)

Special Values of

15

18

Application of the Residue Calculu

The Psi (or

1.4

6 6

The Gaussian

Hypergeometric

f'unction and

58

its Genera l ization

The Gauss Hypergeometric Equation Gauss's

52

Hypergeometric Series

The Hypergeometric Series and Its Analytic Continuation

63 63 64

65

Contents

vi Linear, Quadratic and Cubic Transfom1ations

67

Hypergeometric Representations of Elem en tary Functions Hypcrgcomctric Representations of Other FuncLions

68

The Conftuent Hypergeometric Function

69

67

lmponam Properties of K u m mer s Confluent Hypergeometric '

Function

70

The Generalized

(Gauss

and Kummer)

Hypergeometric

Function

71

Analytk Continuation of the Generalized HypergeomeLric Function

72

Functions Expressible

J .6

Stirling

in Terms of the pFq Function

umbers of the First and Second K i nd

Kind of the Second Kind Relationships Among Stirl i n g Numbers of the Fir t and Second Kind and Bemoulli Numbers 1.7

S tirling Number of the First

76

Stirling Num bers

78

Bernoulli,

Euler and Gcnocchi Polyno mi al s and Numbers

Bernoulli Polynomials and· umbers

The Generalized Bernoulli Polynomial

Euler Polynomial.. and

Relations Between Bemou l l i and Euler Polynomial The Generalized Euler Polynomials and

Numbers

umbers

88 88

Apostol-Bemoulli, Apostol-Euler and Apostol-Genocchi

91

91

Apostoi-Bernoulli Polynomials and Numbers

Apostoi-Genocchi Polynomial

and Number

Remark and Ob ervatio ns Generalizations and Unified Presentations of the

98

Important

99 Apostol Type

Polynomials Inequalities for

100

the Gamma Function and the Double Gamma

Fun��n

105

The Gamma Function and Its Relati es The Double Gamma Fu nction

2

87

90

Polynomials and Numbers

1.9

83 86

of Bernoulli and Euler P oly nom i al s

Gcnocchi Polynomials and

79 81 81

and Number

umbers

Fourier Series Expansions

1.8

73 76

105 112

Problem

112

The Zeta and Related Functions

141

2. I

M u lt iple Hurwitz Zeta Functions

The Analytic Continuation of

Relation hip between Sn The Vardi-Barne

2.2

The Hurwitz

(or

s,

141

s, (s. a)

x) and

B,�l (x)

Multiple Gamma Functions

Generalized) Zeta

Hurwitz's Formula Hermite's Fom1llla

Function

fors( a) for t(s, a) ,

Further I nte gral Representations for s(s. a)

142 150 153 155 156 157 1 ' 59

Comems

vii Some Applications of the A not h er Form for

2.3

Derivative Fom1ula ( 17)

f2(a)

160 162

The Ri c m ann Zcta Function

164 166

Riemann's Functional Equation for �(s)

Relationship between �(s) and the Mathematical

167

Constants B and C

2.4

for� (s) for� (n)

Integral Representations

169

A Summation Identity

172 175

Polylogarithm Functions

176

The Dilogarithm Function

Clausen's Integ ral or

181

Function)

183

The Trilogarithm Function

2.5

The Polylogarithm Functions

185

The L og - Sine Integrals

191

Hurwitz-Lerch Zeta Functions

194

The Taylor Series E xpansi o n of the Lipschitz-Lcrch Transcendent

Evaluation of

L(x, s, a) L(x. -n, a)

198 199

2.6

Generalizations of the Hurwitz-Lerch Zeta Function

2.7

Anal y ti c Continuations of Multiple Zeta Functions Generalized Function

of Gel'fand and Shilov

213

213 220

Euler-Maclaurin Summation Fonnula Problems

3

200

224

Series involving Zctu Functions

245

3 .I

Historical Introduction

245

3.2

Use of the B inom i al

Theorem

247

Applications of Theorems 3.1 and 3.2

3.3

3.4

257

Use of Generating Functions

261

Series Involving Polygamma Func t i o ns

266

Series Involving Polylogarithm Functions

267

Use of Multiple Gamma Functions

269

Evaluation by Using the Gamma !'unction Evaluation in Terms of Catalan'

Constant G

f-urther Evaluation by Using the Triple Gamma !'unction

Applications of Corollary

3.5

3.3

Series

Derivable

Series Derivable

350

from Gauss's Summation Formula 1.4(7) from Kummer's Formula (3) from Other Hypergeometric Summation

Fonnulas Further Summation Fom1ulas Rel a ted to

3.6

339 344 348

Use of Hypergeometric Identities Series Derivable

269

351 354

358

Generalized Ham1onic

Numbers Other Methods and their Applications The Weierstrass Canonical Product Form for the G a m m a Function

361 364 364

Contents

viii

3.7

4

Evaluation by Using Infinite Products

366

Higher-Order D eri v ati v e s of the Gamma Fun c t io n

369

Applications of Series Involving the Zcta Function

375

The Multiple Gamma Functions

375

Matbieu Serie

382

P ro b l e ms

389

Evaluations and Series Representations

399

4.1

399

Evaluation of� 2n) l11e G eneral

Ca

e of

�(2n)

402

4.2

Rapidly Convergent Series for �(2n + I)

4.3

Further Series Representations

4.4

Computational Results

405

Remarks and Obser at ions

5

409 415 422

Problem

433

Determinants of the Lapl acians

445

5.1 5.2

The n-Dimensional Problem

445

Computations Using the Simple and Multiple Gamma

Factorizations Into

Simple and Multiple Gamma Functions

Evaluations of det' !:i,

(n

=

I. 2,

3)

5.3

Computations Using

5.4

Computations using Zeta Regularized Product

5.5

Function,

eries of Zeta Functions

452 457 465

A Lemma on Zeta Regularized Products and a Main Theorem

467

Computations for sm a l l

471

n

Remark s and Observations

472

Problems 6

448 448

473

q Extensions of Some Special Functions and Polynomials

479

6.1

q-Shifted Factorials and q-Binomial Coefficients

479

6.2

q-Derivative, q-Antiderivative and Jack son q-Integral

483

-

q-Deri vative

484

q-Antiderivative and Jackson q-lntegral

484

6.3

q-Binomial Theorem

487

6.4

q-Garnma Function and q-Beta Function

490

q-Gamma

490

Function

495

q-Beta Function

6.5

A q-Extension of the M u l t i pl e Gamma Functions

6.6

q-Bernoulli

q

-

499

Second f3t:q(.:r)

Kind

504

umbers and q-Euler Polynomials

506 509 513

St irl ing

umbers of the

The Polynomial /3* (x)

6.7 6.8 6.9

q-E u ler

497

umbers and q-Bernoulli Polynomials =

!3/>(x; A) of Order 11 The q-Apostoi-Eulcr Polynomials E}">(x; A) of Order 11

The q-Apostol-Bernoulli Polynomials

518

Comems

ix 519

6.10 A Generalized q-Zeta Function

6.1 1

An Auxiliary Func ti o n Defining Generalized q-Zeta Fun ction

519

Application of Eulcr-Maclaurin Summation Fom1tda

524

Multiple q-Zeta Functions Analytic

530

Continuation of gq

and

{q

530

Ana lytic Continuation of M u l tip l e Zeta Functions Special Values of

7

l;q (s1, s2)

533 541

Problems

542

Miscellaneous Result'>

555

7.1

A Set of Useful Mathematical Constants

Euler-Mascherotti

7.2

555

Constant y

555

Series Representations for y

556

A Class of Con stant s Analogous to {Dk}

560

O th er Classes of Mathematical Constants

563

Log-Sine Integrals Involving Series Associated with the Zeta Function and Polylogarithms

568

Analogous Log-Sine Integral

Remark on Cl,(O) and

571

575

GI,(O)

f-urther Rema rk s and Observations

7.3

Applications of the Gamma

578

and Polygamma

Functions Involving

Convolutions of the Rayleigh Functions

581

Se r i e s E x pre ss ib l e in Terms of the 1ft-Function Convolutions 7.4

Bemoulli

of the

582

Rayleigh Functions

and Euler Polynomial at

584

Rational Argument

The Cvijovit-Kiinowsld Summation Formulas Srivastava's Shorter Proofs of Theorem

7.3

and Theorem 7.4

Fonnulas I n v ol v in g the Hurwitz-Lerch Ze ta Function An Application of Lerch's Functional Eq ua t ion

7.5

Closed-Form

Problems

Bibliography

ummation of Trigonometric

erie

587 588

2.5(29)

589 591 593 594

597

603

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Preface

This book is essentially a thoroughly revised, enlarged and updated version of the authors’ work: Series Associated with the Zeta and Related Functions (Kluwer Academic Publishers, Dordrecht, Boston and London, 2001). It aims at presenting a state-of-the-art account of the theories and applications of the various methods and techniques which are used in dealing with many different families of series associated with the Riemann Zeta function and its numerous generalizations and basic (or q-) extensions. Systematic accounts of only some of these methods and techniques, which are widely scattered in journal articles and book chapters, were included in the abovementioned book. In recent years, there has been an increasing interest in problems involving closedform evaluations of (and representations of the Riemann Zeta function at positive integer arguments as) various families of series associated with the Riemann Zeta function ζ (s), the Hurwitz Zeta function ζ (s, a), and their such extensions and generalizations as (for example) Lerch’s transcendent (or the Hurwitz-Lerch Zeta function) 8(z, s, a). Some of these developments have apparently stemmed from an over twocentury-old theorem of Christian Goldbach (1690−1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700−1782), from recent rediscoveries of a fairly rapidly convergent series representation for ζ (3), which is actually contained in a 1772 paper by Leonhard Euler (1707−1783), and from another known series representation for ζ (3), which was used by Roger Ape´ ry (1916−1994) in 1978 in his celebrated proof of the irrationality of ζ (3). This revised, enlarged and updated version of our 2001 book is motivated essentially by the fact that the theories and applications of the various methods and techniques used in dealing with many different families of series associated with the Riemann Zeta function, its aforementioned relatives and its many different basic (or q-) extensions are to be found so far only in widely scattered journal articles published during the last decade or so. Thus, our systematic (and unified) presentation of these results on the evaluation and representation of the various families of Zeta and q-Zeta functions is expected to fill a conspicuous gap in the existing books dealing exclusively with these Zeta and q-Zeta functions. The main objective of this revised, enlarged and updated version is to provide a systematic collection of various families of series associated with the Riemann and Hurwitz Zeta functions, as well as with many other higher transcendental functions, which are closely related to these functions (including especially the q-Zeta and related functions). It, therefore, aims at presenting a state-of-the-art account of the theory and applications of many different methods (which are available in the rather scattered

xii

Preface

literature on this subject, especially since the publication of our aforementioned 2001 book) for the derivation of the types of results considered here. In our attempt to make this book as self-contained as possible within the obvious constraints, we include in Chapter 1 (Introduction and Preliminaries) a reasonably detailed account of such useful functions as the Gamma and Beta functions, the Polygamma and related functions, multiple Gamma functions, the Gauss hypergeometric function and its familiar generalization, the Stirling numbers of the first and second kind, the Bernoulli, Euler and Genocchi polynomials and numbers, the Apostol-Bernoulli, the Apostol-Euler and the Apostol-Genocchi polynomials and numbers, as well as some interesting inequalities for the Gamma function and the double Gamma function. In Chapter 2 (The Zeta and Related Functions), we present the definitions and various potentially useful properties (and characteristics) of the Riemann, Hurwitz and Hurwitz-Lerch Zeta functions and their generalizations, the Polylogarithm and related functions and the multiple Zeta functions, together with their analytic continuations. In Chapter 3 (Series Involving Zeta Functions), we begin by providing a brief historical introduction to the main subject of this book. We then describe and illustrate some of the most effective methods of evaluating series associated with the Zeta and related functions. Further developments on the evaluations and (rapidly convergent) series representations of ζ (s) when s ∈ N \ {1} are presented in Chapter 4 (Evaluations and Series Representations), which also deals with various computational results on this subject. Chapter 5 (Determinants of the Laplacians) considers the problem involving computations of the determinants of the Laplacians for the n-dimensional sphere Sn (n ∈ N). It is here in this chapter that we show how fruitfully some of the series evaluations (which are presented in the earlier chapters) can be applied in the solution of the aforementioned problem. In a brand new Chapter 6 (q-Extensions of Some Special Functions and Polynomials), we first introduce the concepts of the basic (or q-) numbers, the basic (or q-) series and the basic (or q-) polynomials. We then proceed to apply these concepts and present a reasonably detailed theory of the various basic (or q-) extensions of the Gamma and Beta functions, the derivatives, antiderivatives and integrals, the binomial theorem, the multiple Gamma functions, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Apostol-Bernoulli polynomials, the ApostolEuler polynomials and so on. The last chapter (Chapter 7) contains a wide variety of miscellaneous results dealing with (for example) the analysis of several useful mathematical constants, a variety of Log-Sine integrals involving series associated with the Zeta function and Polylogarithms, applications of the Gamma and Polygamma functions involving convolutions of the Rayleigh functions, evaluations of the Bernoulli and Euler polynomials at rational arguments, and the closed-form summation of several classes of trigonometric series. Each chapter in this book begins with a brief outline summarizing the material presented in the chapter and is then divided into a number of sections. Equations in every section are numbered separately. While referring to an equation in another section of

Preface

xiii

the book, we use numbers like 3.2(18) to represent Equation (18) in Section 3.2 (that is, the second section of Chapter 3). At the close of each chapter, we have provided a set of carefully-selected problems, which are based essentially upon the material presented in the chapter. Many of these problems are taken from recent research publications, and (in all such instances) we have chosen to include the precise references for further investigation (if necessary). Another valuable feature of this book is the extensive and up-to-date bibliography on the subject dealt with in the book. Just as its predecessor (that is, the 2001 edition), this book is written primarily as a reference work for various seemingly diverse groups of research workers and other users of series associated with the Zeta and related functions. In particular, teachers, researchers and postgraduate students in the fields of mathematical and applied sciences will find this book especially useful, not only for its detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series associated with the Zeta and related functions, or for its stimulating historical accounts of a large number of problems considered here, but also for its well-classified tables of series (and integrals) and its well-motivated presentation of many sets of closely related problems with their precise bibliographical references (if any).

This page intentionally left blank

Acknowledgements

Many persons have contributed rather significantly to this thoroughly revised, enlarged and updated version, just as to its predecessor (that is, the 2001 edition), both directly and indirectly. Contribution of subject matter is duly acknowledged throughout the text and in the bibliography. Indeed, we are greatly indebted to the various authors whose works we have freely consulted and who occasionally provided invaluable references and advice serving for the enrichment of the matter presented in this book. The first-named author wishes to express his deep sense of gratitude to his wife and colleague, Professor Rekha Srivastava, for her cooperation and support throughout the preparation of this thoroughly revised, enlarged and updated version of the 2001 book. The collaboration of the authors on the 2001 book project was conceptualized as long ago as August 1995, and the preparation of a preliminary outline was initiated in December 1997, during the first-named author’s visits to Dongguk University at Gyeongju. The first drafts of some of the chapters in this book were written during several subsequent visits of the first-named author to Dongguk University at Gyeongju. The final drafts of most of the chapters in the 2001 book were prepared during the second-named author’s visit to the University of Victoria from August 1999 to August 2000, while he was on Study Leave from Dongguk University at Gyeongju. The preparation of this thoroughly revised, enlarged and updated version was carried out, in most part, during the period from January 2008 to January 2009, during the secondnamed author’s visit to the University of Victoria, while he was on Study Leave from Dongguk University at Gyeongju for the second time. Our sincere thanks are due to the appropriate authorities of each of these universities, to the Korea Research Foundation (Support for Faculty Research Abroad under its Research Fund Program) and to the Natural Sciences and Engineering Research Council of Canada, for providing financial support and other facilities for the completion of each of the projects leading eventually to the 2001 edition and this thoroughly revised, enlarged and updated version. We especially acknowledge and appreciate the financial support that was received under the Basic Science Research Program through the National Research Foundation of the Republic of Korea. We take this opportunity to express our thanks to the editorial (and technical) staff of the Elsevier Science Publishers B.V. (especially the Publisher, Ms. Lisa Tickner, for Serials and Elsevier Insights) for their continued interest in this book and for their proficient (and impeccable) handling of its publication. Springer’s permission to publish this thoroughly revised, enlarged and updated edition of the 2001 book is also greatly appreciated. Finally, we should like to record our indebtedness to the members of our respective families for their understanding, cooperation and support throughout this project.

xvi

Acknowledgements

The second-named author and his family would, especially, like to express their appreciation for the first-named author and his family’s hospitality and every prudent consideration during their stay in Victoria for over one year, first from August 1999 to August 2000 and then again from January 2008 to January 2009, while the secondnamed author was on Study Leave from Dongguk University at Gyeongju. H. M. Srivastava University of Victoria Canada Junesang Choi Dongguk University Republic of Korea February 2011

1 Introduction and Preliminaries In this introductory chapter, we present the definitions and notations (and some of the important properties and characteristics) of the various special functions, polynomials and numbers, which are potentially useful in the remainder of the book. The special functions considered here include (for example) the Gamma, Beta and related functions, the Polygamma functions, the multiple Gamma functions, the Gaussian hypergeometric function and the generalized hypergeometric function. We also consider the Stirling numbers of the first and second kind, the Bernoulli, Euler and Genocchi polynomials and numbers and the various families of the generalized Bernoulli, Euler and Genocchi polynomials and numbers. Relevant connections of some of these functions with other special functions and polynomials, which are not listed above, are also presented here.

1.1 Gamma and Beta Functions The Gamma Function The origin of the Gamma function can be traced back to two letters from Leonhard Euler (1707–1783) to Christian Goldbach (1690–1764), just as a simple desire to extend factorials to values between the integers. The first letter (dated October 13, 1729) dealt with the interpolation problem, whereas the second letter (dated January 8, 1730) dealt with integration and tied the two together. The Gamma function 0(z) developed by Euler is usually defined by 0(z) :=

Z∞

e−t tz−1 dt

(<(z) > 0).

(1)

0

We also present here several equivalent forms of the Gamma function 0(z), one by Weierstrass: 0(z) =

 ∞  z −1 z/k e−γ z Y  1+ e z k

(2)

k=1

z ∈ C \ Z− 0;

Z− 0

:= {0, −1, −2, . . .} ,


Zeta and q-Zeta Functions and Associated Series and Integrals. DOI: 10.1016/B978-0-12-385218-2.00001-3 c 2012 Elsevier Inc. All rights reserved.

2

Zeta and q-Zeta Functions and Associated Series and Integrals

where γ denotes the Euler-Mascheroni constant defined by ! n X 1 γ := lim − log n ∼ = 0.57721 56649 01532 86060 6512 . . . , n→∞ k

(3)

k=1

and the other by Gauss:   (n − 1)! nz 0(z) = lim n→∞ z(z + 1) · · · (z + n − 1)   n! (n + 1)z = lim n→∞ z(z + 1) · · · (z + n)   n! nz = lim n→∞ z(z + 1) · · · (z + n)

(4)

(z ∈ C \ Z− 0 ), since lim

n→∞

n nz = 1 = lim . n→∞ (n + 1)z z+n

In terms of the Pochhammer symbol (λ)n defined (for λ ∈ C) by ( 1 (n = 0) (λ)n := λ(λ + 1) · · · (λ + n − 1) (n ∈ N := {1, 2, 3, . . .}),

(5)

the definition (4) can easily be written in an equivalent form: 0(z) = lim

n→∞

(n − 1)! nz (z)n

(z ∈ C \ Z− 0 ).

(6)

By taking the reciprocal of (2) and applying the definition (3), we have #     " n n Y 1 1 1 z  −z/k o = z lim exp 1 + + · · · + − log n z lim 1+ e n→∞ n→∞ 0(z) 2 n k k=1 # "    Y n n 1 1 z  −z/k o = z lim exp 1 + + · · · + − log n z · 1+ e n→∞ 2 n k k=1 ( ) n  Y z −z = z lim n 1+ n→∞ k k=1 "(n−1  )#  ) (Y n  Y 1 −z z 1+ = z lim 1+ n→∞ k k k=1 k=1 ( )   ∞  Y z 1 −z =z 1+ 1+ , k k k=1

Introduction and Preliminaries

3

which yields Euler’s product form of the Gamma function:   ∞  1Y 1 z z −1 0(z) = 1+ 1+ . z k k

(7)

k=1

When t in (1) is replaced by − log t, (1) is also written in an equivalent form: 0(z) =

 Z1  1 z−1 dt log t

(<(z) > 0).

(8)

0

This representation of the Gamma function as well as the symbol 0 are attributed to Legendre. Integration of (1) by parts easily yields the functional relation: 0(z + 1) = z 0(z),

(9)

so that, obviously, 0(z) =

0(z + n) z(z + 1) · · · (z + n − 1)

(n ∈ N0 := N ∪ {0}),

(10)

which enables us to define 0(z) for <(z) > −n(n ∈ N0 ) as an analytic function except for z = 0, −1, −2, . . . , −n + 1. Thus, 0(z) can be continued analytically to the whole complex z-plane except for simple poles at z ∈ Z− 0. The representation (2) in conjunction with the well-known product formula:  ∞  Y z2 sin π z = π z 1− 2 n

(11)

n=1

also yields the following useful relationship between the Gamma and circular functions: 0(z) 0(1 − z) =

π sin πz

(z 6∈ Z := {0, ±1, ±2, . . .}),

(12)

which incidentally provides an immediate analytic continuation of 0(z) from right to the left half of the complex z-plane. Several special values of 0(x), when x is real, are worthy of note. Indeed, from (1) we note that 0(1) =

Z∞ 0

e−t dt = 1,

(13)

4

Zeta and q-Zeta Functions and Associated Series and Integrals

and that 0(x) > 0 for all x in the open interval (0, ∞). Thus, (12) with z = ately yields   √ 1 0 = π, 2

1 2

immedi-

(14)

which, in view of (1), implies Z∞

√ e−t √ dt = π t

(15)

0

or, equivalently, Z∞

√ π exp(−t )dt = . 2 2

(16)

0

By making use of the relation (10), we obtain √   1 (2n)! π 0 n+ = ; 2 22n n!   √ 22n n! 1 0 −n + = (−1)n π ; (n ∈ N0 ). 2 (2n)!

0(n + 1) = n!;

(17)

The last two results in (17) also require the use of (14). The formulas listed under (17) enable us to compute 0(x) when x is a positive integer and when x is half an odd integer, positive or negative. From (2) or (4), it also follows that 0(z) is a meromorphic function on the whole complex z-plane with simple poles at z = −n(n ∈ N0 ) with their respective residues given by Res 0(z) =

z=−n

(−1)n n!

(n ∈ N0 ).

(18)

Since the function 0(1/w) has simple poles at w = −1, − 12 , − 13 , . . . , which implies that w = 0 is an accumulation point of the poles of 0(1/w), the Gamma function 0(z) has an essential singularity at infinity. Furthermore, it follows immediately from (2) that 1/ 0(z) has no poles, and, therefore, 0(z) is never zero.

Pochhammer’s Symbol and the Factorial Function Since (1)n = n!, the Pochhammer symbol (λ)n defined by (5) may be looked upon as a generalization of the elementary factorial; hence, the symbol (λ)n is also referred to as the shifted factorial.

Introduction and Preliminaries

5

In terms of the Gamma function, we have (cf. Definition (5)) (λ)n =

0(λ + n) 0(λ)


λ ∈ C \ Z− 0 ,

(19)

which can easily be verified. Furthermore, the binomial coefficient may now be expressed as   λ λ(λ − 1) · · · (λ − n + 1) (−1)n (−λ)n = = (20) n n! n! or, equivalently, as   λ 0(λ + 1) . = n! 0(λ − n + 1) n

(21)

It follows from (20) and (21) that 0(λ + 1) = (−1)n (−λ)n , 0(λ − n + 1) which, for λ = α − 1, yields 0(α − n) (−1)n = 0(α) (1 − α)n

(α 6∈ Z).

(22)

Equations (19) and (22) suggest the definition: (λ)−n =

(−1)n (1 − λ)n

(n ∈ N; λ 6∈ Z).

(23)

Equation (19) also yields (λ)m+n = (λ)m (λ + m)n ,

(24)

which, in conjunction with (23), gives (λ)n−k =

(−1)k (λ)n (1 − λ − n)k

(0 5 k 5 n).

(25)

For λ = 1, we have (n − k)! =

(−1)k n! (−n)k

(0 5 k 5 n),

which may alternatively be written in the form:  k   (−1) n! (0 5 k 5 n), (−n)k = (n − k)!  0 (k > n).

(26)

(27)

6

Zeta and q-Zeta Functions and Associated Series and Integrals

Multiplication Formulas of Legendre and Gauss In view of the definition (5), it is not difficult to show that (λ)2n = 2

2n



   1 1 1 λ λ+ 2 n 2 2 n

(n ∈ N0 ),

(28)

which follows also from Legendre’s duplication formula for the Gamma function, viz √

π 0(2z) = 2

2z−1



1 0(z) 0 z + 2





 1 3 z 6= 0, − , −1, − , . . . . 2 2

(29)

For every positive integer m, we have (λ)mn = m

mn

 m  Y λ+j−1 j=1

m

(m ∈ N; n ∈ N0 ),

(30)

n

which reduces to (28) when m = 2. Starting from (30) with λ = mz, it can be proved that 0(mz) = (2π) 2 (1−m) mmz− 2 1

1

m Y j=1

  j−1 0 z+ m   1 2 z 6= 0, − , − , . . . ; m ∈ N , m m

(31)

which is known in the literature as Gauss’s multiplication theorem for the Gamma function.

Stirling’s Formula for n! and its Generalizations For a large positive integer n, it naturally becomes tedious to compute n!. An easy way of computing an approximate value of n! for large positive integer n was initiated by Stirling in 1730 and modified subsequently by De Moivre, who showed that n! ∼

 n n √ 2πn e

(n → ∞)

(32)

or, more generally, that 0(x + 1) ∼

 x x √ 2π x e

(x → ∞; x ∈ R),

where e is the base of the natural logarithm.

(33)

Introduction and Preliminaries

7

For a complex number z, we have the following asymptotic expansion:   n   X 1 1 B2k −2n−1 + O z log 0(z) = z − log z − z + log(2π) + 2 2 2k(2k − 1) z2k−1 k=1

(|z| → ∞; | arg(z)| 5 π −  (0 <  < π); n ∈ N0 ), (34) which, upon taking exponentials, yields an asymptotic formula for the Gamma function: r  139 571 2π 1 1 z −z 0(z) = z e − − 1+ + 2 3 z 12 z 288 z 51840 z 2488320 z4   163879 50043869 + + + O z−7 (35) 209018880 z5 75246796800 z6 (|z| → ∞; | arg(z)| 5 π −  (0 <  < π)). The asymptotic formula (35), in conjunction with the recurrence relation (9), is useful in computing the numerical values of 0(z) for large real values of z. Some useful consequences of (34) or (35) include the asymptotic expansions:     1 1 log 0(z + α) = z + α − log z − z + log(2π) + O z−1 2 2 (36) (|z| → ∞; | arg(z)| 5 π − ; | arg(z + α)| 5 π − ; 0 <  < π), and    0(z + α) (α − β)(α + β − 1) α−β =z 1+ + O z−2 0(z + β) 2z (|z| → ∞; | arg(z)| 5 π − ; | arg(z + α)| 5 π − ; 0 <  < π),

(37)

where α and β are bounded complex numbers. Yet another interesting consequence of (35) is the following asymptotic expansion of |0(x + iy)|: √ 1 1 |0(x + iy)| ∼ 2π |y|x− 2 e− 2 π |y| (|x| < ∞; |y| → ∞), (38) where x and y take on real values.

The Beta Function The Beta function B(α, β) is a function of two complex variables α and β, defined by B(α, β) :=

Z1 0

tα−1 (1 − t)β−1 dt = B(β, α)

(<(α) > 0; <(β) > 0)

(39)

8

Zeta and q-Zeta Functions and Associated Series and Integrals

or, equivalently, by Zπ/2 B(α, β) = 2 (sin θ)2α−1 (cos θ)2β−1 dθ

(<(α) > 0; <(β) > 0),

(40)

0

which follows from (39) on setting t = sin 2 θ. The integrals in (39) and (1) are known as the Eulerian integrals of the first and second kind, respectively. Putting t = u/(1 + u) in (39), we obtain the following representation of B(α, β) as an infinite integral: B(α, β) =

Z∞ 0

uα−1 du (<(α) > 0; <(β) > 0). (1 + u)α+β

(41)

The Beta function is closely related to the Gamma function; in fact, we have B(α, β) =

0(α) 0(β) 0(α + β)


α, β 6∈ Z− 0 ,

(42)

which not only confirms the symmetry property in (39), but also continues the Beta function analytically for all complex values of α and β, except when α, β ∈ Z− 0 . Thus, we may write

B(α, β) =

 Z1     α−1  (1 − t)β−1 dt  t 0    0(α) 0(β)    0(α + β)

(<(α) > 0; <(β) > 0) (43)


<(α) < 0; <(β) < 0; α, β 6∈ Z− 0 .

Next we combine the relationship (42) with (40) and (41), and we obtain the following useful integral formulas: Zπ/2 0 sin µ θ cos ν θ dθ = 0



   0 12 ν + 12   2 0 12 µ + 12 ν + 1 1 1 2µ + 2

(<(µ) > −1; <(ν) > −1) (44)

and Z∞ 0

uλ−1 0(λ) 0(µ − λ) du = µ (1 + u) 0(µ)

(0 < <(λ) < <(µ)).

(45)

Introduction and Preliminaries

9

It should be remarked in passing that the integral (44) provides a generalization of Wallis’s formula of elementary calculus and that (45), with µ = 1, yields the familiar infinite integral: Z∞

π uλ−1 du = 0(λ) 0(1 − λ) = 1+u sin πλ

0

(0 < <(λ) < 1),

(46)

which is usually evaluated in the literature by contour integration (see, e.g., Copson [341, p. 139, Example 1]). In addition to (41) and (45), by means of suitable substitutions, a number of definite integrals are expressible in terms of the Beta function: Z1

tα−1 (1 − t)β−1 (1 + at)−α−β dt = (1 + a)−α B(α, β)

0

(a > −1; <(α) > 0; <(β) > 0); Z∞

tβ−1 (1 + at)−α−β dt = a−β B(α, β)

0

(a > 0; <(α) > 0; <(β) > 0); Za

(a > b; <(α) > 0; <(β) > 0);

b

(48)

(t − b)α−1 (a − t)β−1 dt = (a − b)α+β−1 B(α, β)

b

Za

(47)

(t − b)α−1 (a − t)β−1 (a − b)α+β−1 dt = B(α, β) (t − c)α+β (a − c)α (b − c)β

(49)

(50)

(a > b > c; <(α) > 0; <(β) > 0). The following functional equations for the Beta function can be deduced easily from (39) and (42): B(α, β + 1) =

β β B(α + 1, β) = B(α, β); α α+β

B(α, β) B(α + β, γ ) = B(β, γ ) B(β + γ , α) = B(γ , α) B(α + γ , β); B(α, β) B(α + β, γ ) B(α + β + γ , δ) =

0(α) 0(β) 0(γ ) 0(δ) , 0(α + β + γ + δ)

(51) (52) (53)

10

Zeta and q-Zeta Functions and Associated Series and Integrals

or, more generally,   k X 0(α1 ) · · · 0(αn+1 ) (n ∈ N); B αj , αk+1  = 0(α1 + · · · + αn+1 ) j=1 k=1     n+m−1 n+m−1 1 =n (n, m ∈ N). =m B(n, m) n−1 m−1 n Y

(54)

(55)

The Incomplete Gamma Functions The incomplete Gamma function γ (z, α) and its complement 0(z, α) (also known as Prym’s function) are defined by γ (z, α) :=

Zα

tz−1 e−t dt

(<(z) > 0; | arg(α)| < π),

(56)

tz−1 e−t dt

(| arg(α)| < π),

(57)

0

0(z, α) :=

Z∞ α

so that γ (z, α) + 0(z, α) = 0(z).

(58)

For fixed α, 0(z, α) is an entire (integral) function of z, whereas γ (z, α) is a meromorphic function of z, with simple poles at the points z ∈ Z− 0. The following recursion formulas are worthy of note: γ (z + 1, α) = z γ(z, α) − α z e−α ,

(59)

0(z + 1, α) = z 0(z, α) + α e

(60)

z −α

.

The Incomplete Beta Functions The incomplete Beta function Bx (α, β) is defined by Bx (α, β) :=

Zx

tα−1 (1 − t)β−1 dt

(<(α) > 0).

(61)

0

For the associated function: Ix (α, β) =

Bx (α, β) , B(α, β)

(62)

Introduction and Preliminaries

11

we note here the following properties that are easily verifiable: Ix (α, β) = 1 − I1−x (β, α), n   X n j Ix (k, n − k + 1) = x (1 − x)n−j j

(63) (1 ≤ k ≤ n),

(64)

j=k

Ix (α, β) = x Ix (α − 1, β) + (1 − x) Ix (α, β − 1), (α + β − αx) Ix (α, β) = α(1 − x) Ix (α + 1, β − 1) + βIx (α, β + 1), (α + β) Ix (α, β) = α Ix (α + 1, β) + βIx (α, β + 1).

(65) (66) (67)

The Error Functions The error function erf(z), also known as the probability integral 8(z), is defined for any complex z by 2 erf(z) := √ π

Zz

exp(−t2 ) dt = 8(z),

(68)

0

and its complement by 2 erfc(z) := 1 − erf(z) = √ π

Z∞

exp(−t2 ) dt.

(69)

z

Clearly, we have erf(0) = 0

and

erfc(0) = 1,

(70)

and, in view of the well-known result (16), we also have erf(∞) = 1

and

erfc(∞) = 0.

(71)

The following alternative notations: Erf(z) =

√ π erf(z) and 2

Erfc(z) =

√ π erfc(z) 2

(72)

are sometimes used for the error functions. Many authors use the notations Erf(z) and ˆ Erf(z) for the error functions erf(z) and Erf(z), respectively, defined by (68) and (72), ˆ and the notations Erfc(z) and Erfc(z) for their complements. In terms of the incomplete Gamma functions, it is easily verified that     1 1 2 1 1 2 erf(z) = √ γ ,z and erfc(z) = √ 0 , z . (73) 2 2 π π

12

Zeta and q-Zeta Functions and Associated Series and Integrals

The Bohr-Mollerup Theorem We have already observed that Euler’s definition (1) and its such consequences as (9) and (13) enable us to compute all the real values of the Gamma function from the knowledge merely of its values in the interval (0, 1), as noted in conjunction with (35). Since the solution to the interpolation problem is not determined uniquely, it makes sense to add more conditions to the problem. After various trials to find those conditions to guarantee the uniqueness of the Gamma function, in 1922, Bohr and Mollerup were able to show the remarkable fact that the Gamma function is the only function that satisfies the recurrence relationship and is logarithmically convex. The original proof was simplified, several years later, by Emil Artin, and the theorem, together with Artin’s method of proof, now constitute the Bohr-Mollerup-Artin theorem: Theorem 1.1 Let f : R+ → R+ satisfy each of the following properties: (a) log f (x) is a convex function; (b) f (x + 1) = x f (x) for all x ∈ R+ ; (c) f (1) = 1.

Then f (x) = 0(x) for all x ∈ R+ . Instead of giving here the proof of Theorem 1.1 (see Conway [339, p. 179] and Artin [72, p. 14]), we simply state the necessary and sufficient condition for the logarithmic convexity of a given function. Theorem 1.2 Let f : [a, b] → R, and suppose that f (x) > 0 for all x ∈ [a, b] and that f has a continuous second derivative f 00 (x) for x ∈ [a, b]. Then f is logarithmically convex, if and only if  2 f 00 (x) f (x) − f 0 (x) = 0 (x ∈ [a, b]). Remmert [973] admires the following Wielandt’s uniqueness theorem for the Gamma function: It is hardly known that there is also an elegant function theoretic characterization of 0(z). This uniqueness theorem was discovered by Helmut Wielandt in 1939. A function theorist ought to be as much fascinated by Wielandt’s complex-analytic characterization as by the Bohr-Mollerup theorem. For further comment and applications for Wielandt’s theorem, see [675, pp. 47–49], [973], and [1065]. Here, without proof, we present Theorem 1.3 (Wielandt’s Theorem) Let F(z) be an analytic function in the right half plane A := {z ∈ C | <(z) > 0} having the following two properties: (a) F(z + 1) = z F(z) for all z ∈ A; (b) F(z) is bounded in the strip S := {z ∈ C | 1 5 <(z) < 2}.

Then F(z) = a 0(z) in A with a := F(1).

Introduction and Preliminaries

13

1.2 The Euler-Mascheroni Constant γ The divergence of the harmonic series: n ∞ X X 1 1 1 1 = 1 + + + · · · = lim =∞ n→∞ k 2 3 k

(1)

k=1

k=1

was attributed by James Bernoulli to his brother (see [482]). Yet, the connection between 1 + 12 + · · · + 1n and log n was first established in 1735 by Euler [427] (see Walfisz [1203]), who used the notation C for it and stated that it was worthy of serious consideration. We are surprised at Euler’s foresight that there is a huge amount of literature on this famous mathematical constant γ among which we just refer to the book [543] and the references therein. The Euler (or, more precisely, the Euler-Mascheroni) constant γ is defined as follows (see Eq. 1.1(3)): γ : = lim (Hn − log n) n→∞

∼ = 0.57721 56649 01532 86060 6512 09008 24024 31042 · · · ,

(2)

where Hn are called harmonic numbers defined by Hn :=

n X 1 k=1

k

(n ∈ N).

(3)

The symbol γ was first used by Mascheroni in 1790 (see [802]) and the notation C has still been used for the notation γ in (2) (see, e.g., [505]). In fact, an := Hn − log n (n ∈ N) is a decreasing sequence and 0 < an < 1 for all n ∈ N. Thus, the convergence of the sequence in (2) follows by the monotone convergence theorem (see, e.g., [1202, p. 45]). It is noted that γ is a constant so chosen that 0(1) = 1 in the Weierstrass product form 1.1(2) of the Gamma function 0(z), and the constant γ is the very Euler constant in (2). The relatively more familiar constants are π and e, whose transcendence was shown by Ferdinand Lindemann in 1882 and Charles Hermite in 1873, respectively. The true nature of the Euler constant γ (whether an algebraic or a transcendental number) has not yet been known. This was a part of the famous Hilbert’s seventh problem. David Hilbert [557] announced 23 problems for the twentieth century in the Second International Congress of Mathematicians at Paris in 1900. We introduce here only his seventh problem: Irrationality and Transcendence of Certain Numbers. G. H. Hardy was alleged to have offered to give up his Savilian Chair at Oxford to anyone who proved γ to be irrational (see [543, p. 52]). The degree of possible rationality of γ has been tried (see [1222], [543, p. 97]). Appell [69] gave a proof that γ is irrational. Appell himself’s finding an error, quickly he published a retraction, within a couple of weeks, of his original announcement (see Ayoub [82]).

14

Zeta and q-Zeta Functions and Associated Series and Integrals

Euler [427] gave the formula 1+

1 1 B2 B4 B6 1 + · · · + = γ + log x + − − − + ··· , 2 x 2x 2x2 4x4 6x6

Bn being Bernoulli numbers, in which, by putting x = 10, he calculated γ = 0.57721 56649 01532 5 · · · . Mascheroni [802] evaluated the value of Euler constant with 32 figures as follows: γ = 0.57721 56649 01532 86061 811209008239 · · · Soldner (see [482]) computed the value of γ as γ = 0.57721 56649 01532 86060 6065 · · · , which differs from Mascheroni’s value in the twentieth place. In fact, Mascheroni’s value turned out to be incorrect. Nonetheless, since Mascheroni’s error has led to eight additional calculations of this celebrated mathematical constant, so γ is often called the Euler-Mascheroni constant. Knuth [678] computed the first 1271 decimals. Gourdon and Demichel [503] computed a record 108 million digits of γ . Kondo [693] has computed γ to 2 billion digits, which is apparently the current world record. The Euler (or, more popularly, the Euler-Mascheroni) constant γ is considered the third important mathematical constant next to π and e. The mathematical constants π , e and γ are often referred to as the holy trinity. The constant γ has been involved in a variety of mathematical formulas and results. For instance, the book [505] contains about 160 formulas involving γ . Conversely, Wilf [1228] posed as a problem the following elegant infinite product formula, which contains all of the three most important mathematical constants π , e and γ (see Eq. 3.6(19)): ∞  Y

− 1j

e

j=1



1 1 1+ + 2 j 2j



π

π

e 2 + e− 2 = . π eγ

(4)

Choi et al. [275] presented several general infinite product formulas, which include, as their special cases, the product formula (4) of Wilf [1228] and other product formulas given by Choi and Seo [285]. The function d(n) is the number of divisors of n ∈ N, including 1 and n. The average order of d(n) was proved by Dirichlet in 1838 that   n 1X 1 d(k) = ln n + 2γ − 1 + O √ n n

(n → ∞),

(5)

k=1

where γ is the Euler-Mascheroni constant (see [538, p. 264,Theorem 320]). Very recently, Pillichshammer [893] treated this subject in a more general way. A lot of

Introduction and Preliminaries

15

series representations of γ have been presented (see, e.g., [12, 208, 264, 274, 286, 480, 531, 963, 964, 1094, 1179]), such as (see Eq. 3.4(23)) γ=

∞ X

(−1)k

k=2

ζ (k) , k

(6)

where ζ (s) is the Riemann Zeta function given in 2.3(1). Sondow [1051] presented an elegant double integral representation:

γ=

Z1 Z1

x−1 dx dy. (1 − xy) ln(xy)

0 0

(7)

In fact, many integral representations of γ have been developed. Here, in what follows, several further integral representations of γ with mild generalizations of some known formulas of γ are presented.

A Set of Known Integral Representations for γ Among a variety of known integral representations for γ , here we choose to recall some of them (see, e.g., [53, 54, 220, 264, 284, 365, 505, 1094, 1225]):

γ=

Z1

1 − e−x dx − x

0

e−x dx. x

(8)

1

 Z∞ −x γ= e 0

γ = 1+

1 1 − −x 1−e x

Z∞  0

γ=

Z∞

Z1



1 e−x − 1 + x 1+x 1

1 − e−x − e− x x

(9)

dx. 

dx . x

(10)

! dx.

(11)

0

γ=

Z∞ 0

γ =2

 1 dx − e−x . 1+x x

Z∞ 0

1 2 − e−x 1 + x2



(12)

dx . x

(13)

16

Zeta and q-Zeta Functions and Associated Series and Integrals

γ=

Z∞ 0

 1 −x dx −e . x 1 + x2

(14)

Z∞  dx 2 γ =2 e−x − e−x . x

(15)

0

Z∞

4 γ= 3

4

 dx

.

(16)

Z∞  2 dx 4 γ =4 e−x − e−x . x

(17)

e−x − e−x

x

0

0

1 γ= 1 − 2−n γ = 2n

Z∞h   i dx n exp −x2 − e−x x

(18)

0

Z∞

1 n+1 1 + x2

0

(n ∈ N).

− e−x

2



dx x

(n ∈ Z),

(19)

where Z denotes the set of integers.

γ =2

n

Z∞ 0

γ=

Z∞  0

γ=

Z∞ 0

γ = 1+

 dx  1 2n − exp −x x 1 + x2

(20)

 dx 1 − cos x . 1+x x

(21)

 1 dx − cos x . x 1 + x2

(22)

Z∞ 0

γ =2

(n ∈ N).

Z∞ 

 sin x dx 1 − . 1+x x x 2

e−x − cos x

 dx x

(23)

.

(24)

0

γ = − ln p +

Z∞  0

 2 dx arccot x − e−px π x

(p > 0).

(25)

Introduction and Preliminaries

Z∞ 

3 γ = +2 2

0

γ=

17

cos x − 1 1 + 2 2(1 + x) x



dx . x

(26)

Zπ/2

h i dx . 1 − sec 2 x cos (tan x) tan x

(27)

0

2 γ = − ln p − π

Z∞

sin (p x) ln x

dx x

(p > 0).

(28)

0

2 γ = 1 − ln(2p) − pπ

Z∞ 

sin 2 (px) x2

 ln x dx

(p > 0).

(29)

0



q

1 q γ = 1+ ln p p−q p

2 + π(p − q)

Z∞ 

cos (px) − cos (qx) x2

 ln x dx

0

(30)

(p > 0; q > 0; p 6= q). Z∞

1 γ = +2 2

0

γ=

Z1  x− 0

γ =−

Z∞

x dx . 2 2π 1+x e x −1

1 1 − log x



dx . x log x Z1

−x

0

(32)

  1 dx. log log x

(33)

  π x   sin π xu − x dudx. 2 sin π u

(34)

log x dx = −

e

(31)

0 1

γ = log 2 − π

Z1 Z 2 tan 0 0

γ=

1 +2 2

Z∞

sin (tan−1 x) dx.
√ e2π x − 1 1 + x2

0

γ = log 2 − 2

Z∞ 0

γ = 1+

Z∞  0

sin (tan−1 x) dx.
√ e2π x + 1 1 + x2

sin x cos x − x



log x dx. x

(35)

(36)

(37)

18

Zeta and q-Zeta Functions and Associated Series and Integrals

γ=

Z∞

 dx 1 − cos x . 2 x

(38)

0

1 B2 B4 B2n γ= + + + ··· + − (2n + 1)! 2 2 4 2n

Z∞

Q2n+1 (x) dx, x2n+2

(39)

1

where the functions Qn (x) are defined by  1  (n = 1; 0 < x < 1), x − 2 Qn (x) :=   1 B (x − [x]) (n ∈ N \ {1}; 0 5 x < ∞), n n! Bn := Bn (0) and Bn (x) being the Bernoulli numbers and polynomials, respectively (see [1094, Section 1.6]). As observed by Knopp [676] by an explicit example with n = 3 in (2.61), the approximate value of γ can easily be calculated with much greater accuracy than before (and, theoretically, to any degree of accuracy whatever) by means of the formula (39).

Further Integral Representations for γ Very recently, Choi and Srivastava [302] presented several further integral representations for γ by making use of some formulas in the previous subsection and other known formulas for log 0(z), ψ(z) (Section 1.3) and the Hurwitz (or generalized) Zeta function ζ (s, a) (Section 2.2) or the Riemann Zeta function ζ (s) (Section 2.3) in conjunction with the residue calculus. Here, we choose to record some of them: We begin by recalling an integral formula for log z (see [1225, p. 248]). The following integral formula holds true for log z: Z∞

−t

e

−t z

−e


dt = t

0

Z1   dt = log z tz−1 − 1 log t


<(z) > 0 ,

(40)

0

where the log z is an appropriate branch of the multiple-valued function log z, such as log z = ln |z| + i arg z

|z| > 0; α < arg(z) < α + 2π

for some real α ∈ R with possibly −π 5 α 5 − π2 .




Introduction and Preliminaries

19

If (40) is used in the formulas (25), (28), (29) and (30) and tan x is replaced by x in (27), the following integral formulas for γ are obtained: Each of the following integral representations holds true for γ : γ=

Z∞  0

2 γ =− π

 dx 2 arccot x − e−x , π x

(41)

 Z∞  π dx 2 e−x − e−p x + ln x sin (p x) x

(p > 0),

(42)

0

2 γ = 1− pπ

 Z∞  h i pπ sin 2 (px) dx −x 2 e − exp −(2p) x + ln x x x

(p > 0),

(43)

0

2 γ = 1+ π(p − q)

Z∞

−x

e

"  π # ! qq 2 cos (px) − cos (qx) dx − exp − p x + ln x p x x

0

(p > 0; q > 0)

(44)

and Z∞ h i dx γ= cos 2 (arctan x) − cos x . x

(45)

0

If x is replaced by xp in (10), (12), (21), (23) and (26), the following mildly more general formulas for γ are obtained. Each of the following integral representations holds true for γ : γ =p

Z∞ 0

γ =p

Z∞  0

γ = 1+p


dx 1 p − exp −x 1 + xp x 
dx 1 p − cos x 1 + xp x

Z∞  0

γ = 1+p

Z∞ 0

(p > 0),

(p > 0),

1 exp (−x p ) − 1 + 1 + xp xp 1 sin (x p ) − 1 + xp xp



(46)

dx x



dx x

(47)

(p > 0),

(p > 0)

(48)

(49)

20

Zeta and q-Zeta Functions and Associated Series and Integrals

and 3 γ = + 2p 2

Z∞  0

 1 cos (x p ) − 1 dx + p 2p 2(1 + x ) x x

(p > 0).

(50)

It is noted that the case p = 2 of (46) would obviously reduce to (13). A class of vanishing integrals is provided just below. The following vanishing integral formula holds true: Z∞  0

  Z∞  q−1 1 dx x − xp−1 1 − = dx = 0 1 + x p 1 + xq x (1 + x p ) (1 + xq )

(51)

0

(p > 0; q > 0). Proof. The integral (51) is separated into two parts as follows:  1 ∞   Z∞  q−1 Z Z  q−1 x − xp−1 x − xp−1   dx = + dx, (1 + x p ) (1 + xq ) (1 + x p ) (1 + xq ) 0

0

1

which, upon replacing x by 1/x in the second integral, is seen to vanish to 0.



Applying Eq. (51) to Eqs. (46)–(50), Choi and Srivastava [302] derived much more general integral representations for γ , which are recorded here. γ =p

Z∞


dx 1 p − exp −x 1 + xq x

0

γ =p

Z∞  0

γ = 1+p


dx 1 p − cos x 1 + xq x

Z∞  0

γ = 1+p

Z∞ 0

3 γ = + 2p 2

0

γ=

pq q−p

0



(p > 0; q > 0),

 1 exp (−x p ) − 1 dx + 1 + xq xp x  1 sin (x p ) dx − 1 + xq xp x

Z∞ 

Z∞

(p > 0; q > 0),

(p > 0; q > 0),

(p > 0; q > 0),

 cos (x p ) − 1 1 dx + q 2p 2(1 + x ) x x



 dx exp −xq − exp −x p x

(p > 0; q > 0),

(p > 0; q > 0; p 6= q),

(52)

(53)

(54)

(55)

(56)

(57)

Introduction and Preliminaries

pq γ= q−p

Z∞



21



 dx cos xq − cos x p x

(p > 0; q > 0; p 6= q),

(58)

dx x

(59)

0

pq γ = 1+ q−p 3 2pq γ= + 2 q−p

Z∞ 

sin (xq ) sin (x p ) − xq xp



(p > 0; q > 0; p 6= q),

0

Z∞ 

cos (x p ) − 1 cos (xq ) − 1 − x2p x2q



dx x

(60)

0

(p > 0; q > 0; p 6= q), γ=

pq q−p

Z∞



cos (x p ) − exp(−x p )

 dx x

(p > 0; q > 0; p 6= q),

(61)

0

p pq γ= + p−q p−q

Z∞ 

exp(−x p ) −

sin (x p ) xp



dx x

(p > 0; q > 0; p 6= q),

0

(62) 3p pq γ= + 2(p − q) p − q

Z∞ 

exp(−x p ) +

2 [cos (x p ) − 1] x2q



dx x

(63)

0

(p > 0; q > 0; p 6= q), p pq γ= + p−q p−q

Z∞ 

cos (x p ) −

sin (x p ) xp



dx x

(p > 0; q > 0; p 6= q), (64)

0

3p pq γ= + 2(p − q) p − q

Z∞ 

2 [cos (x p ) − 1] cos (x ) + x2q p



dx x

0

(65)

(p > 0; q > 0; p 6= q), 3p − 2q pq γ= + 2(p − q) p − q

Z∞ 

sin (x p ) 2 [cos (x p ) − 1] + xp x2q



dx x

0

(66)

(p > 0; q > 0; p 6= q), p pq γ= + p−q p−q

Z∞ 

 exp (−x p ) − 1 dx p + exp(−x ) p x x

0

(p > 0; q > 0; p 6= q),

(67)

22

Zeta and q-Zeta Functions and Associated Series and Integrals

pq p + γ= p−q p−q

Z∞ 


dx exp (−x p ) − 1 p + cos x xp x

(68)

0

(p > 0; q > 0; p 6= q), pq γ = 1+ p−q

Z∞ 

exp (−x p ) − 1 sin (x p ) + xp xp



dx x

(69)

0

(p > 0; q > 0; p 6= q), pq 2p − 3q + γ= 2(p − q) p − q

Z∞ 

exp (−x p ) − 1 2 [1 − cos (x p )] + xp x2p



dx x

0

(70)

(p > 0; q > 0; p 6= q). It is noted that the integral formula (57) is recorded in [505, p. 364, Entry 3.476-2] and many (if not all) of the integral formulas in the previous subsection can be seen to be special cases of the corresponding integral formulas asserted in this subsection.

From an Application of the Residue Calculus Consider a function f (z) given by f (z) =

1 z



1 n − ei z 1 + zn



(n ∈ N).

Since lim f (z) =

z→0

(n = 1)

  −1 − i 

(n ∈ N \ {1}),

0

the function f (z) has a removable singularity at z = 0 and simple poles at (2k + 1)πi z = exp n 



(k = 0, 1, . . . , n − 1).

We now consider a counterclockwise-oriented simple closed contour: C := Cδ ∪ L1 ∪ CR ∪ L2

(0 < δ < 1 < R),

where Cδ : z = δ eiθ

  π θ varies from to 0 , 2n

Introduction and Preliminaries

23

L1 a line segment from δ to R on the positive real axis,  π θ varies from 0 to 2n

CR : z = R eiθ and  L2 : z = x exp

iπ 2n



(x varies from R to δ) ,

π that is, a line segment on the half-line beginning at the origin with the argument 2n . Since f (z) is analytic throughout the domain interior to and on the closed contour C, it follows from the Cauchy-Goursat theorem that

  Z Z Z Z    + + +  f (z) dz = 0, L1

Cδ

CR

L2

which, upon taking the limits as δ → 0+

and R → ∞

and equating the real and imaginary parts of the last resulting equation, yields the following two interesting integral identities: Z∞  0

  Z∞ 
dx
dx 1 1 n n = − cos x − exp −x 1 + xn x x 1 + x2n

(n ∈ N)

(71)

0

and Z∞ 0

xn−1 dx = 1 + x2n

Z∞

π sin (xn ) dx = x 2n

(n ∈ N).

(72)

0

It is noted that the integral identity (71) is a special case of (52) or (53). Moreover, (72) can be evaluated, as above, by applying the residue calculus to another function f(z) =

exp (izn ) z

(n ∈ N)

and a counterclockwise-oriented simple closed contour C := Cδ ∪ L1 ∪ CR ∪ L2

(0 < δ < R),

24

Zeta and q-Zeta Functions and Associated Series and Integrals

where Cδ and L1 are the same as above, 

CR : z = R eiθ

θ varies from 0 to

π n

and  L2 : z = x exp

iπ n



(x varies from R to δ).

We conclude this section by remarking that more integral representations for γ can be obtained by applying the same techniques employed here (see [302]) or other methods (if any) to some other known formulas that have not been used (see [572]).

1.3 Polygamma Functions The Psi (or Digamma) Function The Psi (or Digamma) function ψ(z) defined by d 0 0 (z) ψ(z) := {log 0(z)} = dz 0(z)

or

log 0(z) =

Zz

ψ(t)dt

(1)

1

possesses the following properties: ψ(z) = lim

n→∞

ψ(z) = −γ −

log n −

n X k=0

1 + z

∞ X n=1

= −γ + (z − 1)

! 1 ; z+k

z n(z + n)

∞ X n=0

1 , (n + 1)(z + n)

(2)

(3)

where γ is the Euler-Mascheroni constant defined by 1.1(3) (or 1.2(2)). These results clearly imply that ψ(z) is meromorphic (that is, analytic everywhere in the bounded complex z–plane, except for poles) with simple poles at z = −n(n ∈ N0 ) with its residue −1. Also we have ψ(1) = −γ ,

(4)

which follows at once from (3). It is noted that, very recently, Bagby [83] proved (4) in another way.

Introduction and Preliminaries

25

The following additional properties of ψ(z) can be deduced from known results for 0(z) : log

  ∞  X 0(z + 1) 1 1 = log z = −γ + − log 1 + ; 0(z) n+1 n+z n=0   ∞  X 1 1 − log 1 + , ψ(z) = log z − n+z n+z

(5)

(6)

n=0

which follows from (3) and (5); ψ(z + n) = ψ(z) +

n X k=1

1 z+k−1

(n ∈ N);

(7)

1 ψ(z) − ψ(−z) = −π cot π z − ; z 1 ψ(1 + z) − ψ(1 − z) = − πcot πz; z ψ(z) − ψ(1 − z) = −π cot π z;     1 1 +z −ψ − z = π tan π z; ψ 2 2  m−1  1 X k ψ(mz) = log m + ψ z+ (m ∈ N). m m

(8) (9) (10) (11) (12)

k=0

Integral Representations for ψ(z) Expanding (1 − t)−1 into a series, integrating term by term and using (3), we get Z1   ψ(z) = −γ + 1 − tz−1 (1 − t)−1 dt

(<(z) > 0),

(13)

0

which, upon replacing t by e−t , yields ψ(z) = −γ +

Z∞



−1 e−t − e−tz 1 − e−t dt

(<(z) > 0).

(14)

0

Making use of (10), it follows from (13) and (14) that ψ(z) = −γ − πcot πz +

Z1 0


1 − t−z (1 − t)−1 dt

(<(z) < 1)

(15)

26

Zeta and q-Zeta Functions and Associated Series and Integrals

and ψ(z) = −γ − π cot π z +

Z∞

1 − etz





−1 et − 1 dt

(<(z) < 1).

(16)

0

It is not difficult to derive each of the following integral representations for ψ(z). Z∞  −1 −t
−1 −tz  e dt ψ(z) = t e − 1 − e−t

(<(z) > 0),

(17)

0

which is due to Gauss and reduces, when z = 1, to the integral expression 1.2(9) for the Euler-Mascheroni constant γ ; ψ(z) =

Z∞ 

 e−t − (1 + t)−z t−1 dt

(<(z) > 0),

(18)

0

which is due to Dirichlet and reduces, when z = 1, to 1.2(11); Z∞h i ψ(z) = −γ + (1 + t)−1 − (1 + t)−z t−1 dt

(<(z) > 0);

(19)

0

Z∞ h
−1 i −tz e dt ψ(z) = log z + t−1 − 1 − e−t

(<(z) > 0),

(20)

0

1 ψ(z) = log z − − 2z

Z∞ 

1 − e−t


−1

− t−1 −

 1 −tz e 2

(<(z) > 0),

(21)

0

ψ(z) = log z +

Z∞h

1 − e−t


−1

i + t−1 − 1 e−tz dt

(<(z) > 0),

(22)

0

and 1 ψ(z) = log z − − 2z

Z∞ 

t

e −1


−1

−t

−1

 1 −tz + e dt 2

(<(z) > 0),

(23)

(<(z) > 0),

(24)

0

all four being attributed to Binet; 1 ψ(z) = log z − − 2 2z

Z∞  −1  −1 t 2 + z2 e2πt − 1 t dt 0

which, in the special case when z = 1, yields 1.2(31).

Introduction and Preliminaries

27

Gauss’s integral representation (17) also yields Malmste´ n’s formula:

log 0(z) =

Zz

ψ(t) dt

1

Z∞ " = 0

(25) 1 − e−(z−1)t (z − 1) − 1 − e−t

#

e−t dt t

(<(z) > 0).

From (23), we can deduce that    −tz Z∞ 1 1 1 1 e log 0(z) = z − log z − z + 1 + − + t dt 2 2 t e −1 t 0

Z∞  − 0

1 1 1 − + t 2 t e −1



e−t dt t

(26) (<(z) > 0).

The second integral in (26) can readily be evaluated: Z∞  0

1 1 1 − + 2 t et − 1



1 e−t dt = 1 − log(2π) t 2

(27)

by using an elementary (yet technical) separation and recalling a formula for log z: Z∞ log z =

e−t − e−zt


dt t

(<(z) > 0).

(28)

0

Now, putting (27) into (26) yields Binet’s first expression for log 0(z):   1 1 log 0(z) = z − log z − z + log (2π) 2 2 ∞  −zt Z  1 1 1 e + − + t dt 2 t e −1 t

(29) (<(z) > 0),

0

which may also give an approximate expression for log 0(z), just as the special case of 1.1(34), when n = 0.

28

Zeta and q-Zeta Functions and Associated Series and Integrals

Binet’s second expression for log 0(z) may be written as follows:   1 1 log 0(z) = z − log z − z + log(2π) 2 2 Z∞ arctan (t/z) +2 dt (<(z) > 0). e2π t − 1

(30)

0

From 1.1(12) and (25), we obtain Kummer’s expression for log 0(z): log 0(z) =

1 1 log π − log(sin π z) 2 2 h  i   Z∞ sinh 1 − z t 2 dt 1  −t   − (1 − 2z) e  + 1 2 t sinh t

(31) (0 < <(z) < 1).

2

0

By applying (31), Kummer derived the following Fourier series for log 0(x): ∞

log 0(x) =

X 1 1 log π − log(sin π x) + 2 an sin (2nπx) 2 2

(0 < x < 1),

(32)

n=1

where Z∞ 

 e−t dt 2nπ − an = t2 + 4n2 π 2 2nπ t 0 ∞   Z  Z∞ −t Z∞
1  dt dt 1 e − e−2π nt  = − cos t + dt + cos t − e−t 2nπ t t t 1 + t2 0

0

0

1 (γ + log 2π + log n). = 2nπ

(33)

Since log(sin πx) = − log 2 −

∞ X 1 cos (2πnx) n

(0 < x < 1),

(34)

n=1

by rewriting (32) in the form:   1 1 log 0 (x) = − x (γ + log 2) + (1 − x) log π − log(sin πx) 2 2 ∞ X log n + sin(2πnx) (0 < x < 1), πn n=1

(35)

Introduction and Preliminaries

29

we finally obtain log 0(x) =

1 log (2π) 2  ∞  X γ + log(2πn) 1 cos (2πnx) + sin(2πnx) + 2n πn

(0 < x < 1).

n=1

(36) The Fourier series (36) readily implies each of the integral formulas: Z1

sin(2πnx) log 0(x) dx =

γ + log(2πn) 2π n

cos(2πnx)log 0(x) dx =

1 4n

(n ∈ N),

(37)

0

Z1

(n ∈ N),

(38)

0

and Z1

log 0(x) dx =

1 log(2π). 2

(39)

0

Considering 1.1(31) and 1.1(34), when n = 0, we readily obtain a more general integral formula than (39) above: Zx+1 1 log 0(t) dt = x log x − x + log(2π) 2

(x = 0),

(40)

x

by observing the following relation:

lim

m→∞

m−1 X k=0

  Z1 Zx+1 1 k log 0 x + = log 0(x + u) du = log 0(t) dt. m m x

0

The formula (40) can
be extended, without difficulty, to the case, when x ∈ C \ Z− Z− := {−1, −2, . . .} and the principal values of the involved logarithms are taken, and can also be further generalized (by iteration) as follows: Zx+n n−1 X 1 log 0(t) dt = (x + k) log(x + k) − nx − n(n − 1) 2 x

k=0

1 + n log(2π) 2

(41) (n ∈ N).

30

Zeta and q-Zeta Functions and Associated Series and Integrals

  Gauss’s Formulas for ψ pq Taking z =

p q

(0 < p < q; p, q ∈ N) in (13) and t = x p , we obtain

  Z1 q−1 p x − xp−1 γ +ψ =q dx, q 1 − xp

(42)

0

which, upon noting that q−1

p

xp−1 − xq−1 X ωk − 1 q = xp − 1 x − ωk k=1

   2πik ωk := exp , q

(43)

readily yields

γ +ψ

     q−1 q−1  p 2kpπ X 2kpπ π X kπ (2k − q) sin 1 − cos = − log 2 sin . q 2q q q q k=1

k=1

(44) By applying the known trigonometric identities: q−1 X

cos 2kα =

1 sin (2q − 1)α + 2 2 sin α

(45)

sin 2kα =

cos α − cos (2q − 1)α 2 sin α

(46)

k=0

and q−1 X k=1

in (44), we obtain Gauss’s formula:     q−1 X p π pπ 2kpπ kπ ψ = − γ − cot − log q + cos log 2 sin q 2 q q q k=1

(47)

(0 < p < q; p, q ∈ N), which implies that, for a positive proper fraction z, the value of ψ(z) can be expressed as a finite combination of elementary functions, and, yet, by means of 1.1(10), may be extended to every rational value of z.

Introduction and Preliminaries

31

By reversing the order of summation, Gauss’s formula (47) can be rewritten in the following form:   π pπ p = − γ − cot − log q ψ q 2 q     [(q−1)/2] X kπ 2kpπ log 2 sin + rq (p), +2 cos q q

(48)

k=1

where [x] denotes the greatest integer ≤ x, and rq (p) =



(−1)p log 2 0

if q is even, if q is odd.

If we rewrite (42) as

γ −q

Z1 0

  Z1 p−1 xq−1 x p dx = −ψ −q dx, p 1−x q 1 − xp 0

where the left-hand side is independent of p and only the right-hand side is dependent on p, we can derive Gauss’s second formula: q X p=1

ψ

       2pkπi 2kπi p exp = q log 1 − exp q q q

(q ∈ N; k ∈ Z).

(49)

Special Values of ψ(z) Setting z = 1 and z = ψ(n) = −γ +

1 2

in (7), we obtain

n−1 X 1 k=1

k

(n ∈ N)

(50)

and   n−1 X 1 1 ψ n+ = −γ − 2 log 2 + 2 2 2k + 1

(n ∈ N0 ),

(51)

k=0

it being understood (here as well as throughout this book) that an empty sum is nil. By suitably applying the various formulas for the ψ–function, we [1094, pp. 20–22] could derive the following special values of ψ(z), some of which were corrected and

32

Zeta and q-Zeta Functions and Associated Series and Integrals

simplified by Tee [1146] as follows:   1 ψ = −γ − 2 log 2; 2   3 1 1 √ ψ = −γ − π 3 − log 3; 3 6 2   1 √ 3 2 = −γ + π 3 − log 3; ψ 3 6 2   1 1 = −γ − π − 3 log 2; ψ 4 2   1 3 ψ = −γ + π − 3 log 2; 4 2 r √ √   π 2√ 5 5 1+ 5 1 = −γ − 1+ 5 − log 5 − log ; ψ 5 2 5 4 2 2 r √ √   π 2√ 5 5 1+ 5 2 ψ = −γ − 1− 5 − log 5 + log ; 5 2 5 4 2 2 r √ √   π 2√ 5 5 1+ 5 3 ψ = −γ + 1− 5 − log 5 + log ; 5 2 5 4 2 2 r √ √   π 2√ 5 1+ 5 5 4 = −γ + 1+ 5 − log 5 − log ; ψ 5 2 5 4 2 2   1 √ 3 1 = −γ − π 3 − log 3 − 2 log 2; ψ 6 2 2   5 3 1 √ ψ = −γ + π 3 − log 3 − 2 log 2; 6 2 2   √ √ 1 π √ ψ = −γ − ( 2 + 1) − 4 log 2 − 2 log( 2 + 1); 8 2   √ √ 3 π √ ψ = −γ − ( 2 − 1) − 4 log 2 + 2 log( 2 + 1); 8 2   √ √ 5 π √ ψ = −γ + ( 2 − 1) − 4 log 2 + 2 log( 2 + 1); 8 2   √ √ 7 π √ ψ = −γ + ( 2 + 1) − 4 log 2 − 2 log( 2 + 1); 8 2 √ q    √ √  π 1 5 5 = −γ − log 2 + 5 ; ψ 5 + 2 5 − 2 log 2 − log 5 − 10 2 4 2 √ q    √ √  3 π 5 5 ψ = −γ − 5 − 2 5 − 2 log 2 − log 5 + log 2 + 5 ; 10 2 4 2 √ q    √ √  7 π 5 5 ψ 5 − 2 5 − 2 log 2 − log 5 + = −γ + log 2 + 5 ; 10 2 4 2

Introduction and Preliminaries

33

√ q    √ √  9 π 5 5 ψ = −γ + 5 + 2 5 − 2 log 2 − log 5 − log 2 + 5 ; 10 2 4 2   √ √ √ 3 1 π ψ = −γ − (2 + 3) − 3 log(2 + 3) − log 3 − 3 log 2; 12 2 2   √ √ √ π 5 3 = −γ − (2 − 3) + 3 log(2 + 3) − log 3 − 3 log 2; ψ 12 2 2   √ √ √ π 3 7 ψ = −γ + (2 − 3) + 3 log(2 + 3) − log 3 − 3 log 2; 12 2 2   √ √ √ 11 π 3 ψ = −γ + (2 + 3) − 3 log(2 + 3) − log 3 − 3 log 2; 12 2 2   1 ψ − = 2 − γ − 2 log 2; 2 √   1 3 3 ψ − = 3−γ + π − log 3; 3 6 2 √   6 3 3 5 = −γ − π − log 3 − 2 log 2. ψ − 6 5 2 2

The Polygamma Functions The Polygamma functions ψ (n) (z) (n ∈ N) are defined by ψ (n) (z) :=

dn+1 dn log 0(z) = n ψ(z) n+1 dz dz

(n ∈ N0 ; z 6∈ Z− 0 ).

(52)

In terms of the generalized (or Hurwitz) Zeta function ζ (s, a) (see Section 2.2), we can write ψ

(n)

(z) = (−1)

n+1

n!

∞ X k=0

1 = (−1)n+1 n! ζ (n + 1, z) (k + z)n+1

(n ∈ N; z 6∈ Z− 0 ), (53)

which may be used to deduce the properties of ψ (n) (z) (n ∈ N) from those of ζ (s, z) (s = n + 1; n ∈ N). It is also easy to have the following expressions: ψ (n) (z + m) − ψ (n) (z) = (−1)n n!

m X k=1

1 (z + k − 1)n+1

(m, n ∈ N0 )

(54)

and ψ (n) (z) − (−1)n ψ (n) (1 − z) = −π

dn {cot πz} dzn

(n ∈ N0 ),

(55)

34

Zeta and q-Zeta Functions and Associated Series and Integrals

it being understood (as usual) that ψ (0) (z) = ψ(z). By using Gauss’s multiplication formula for 0(z) in 1.1(31), it is easy to get the multiplication formula for ψ(n) (z):   m X j−1 (n, m ∈ N). (56) ψ(n) (m z) = ψ(n) z + m j=1

Davis [368] extended Gauss’s result (47) (or (48)) to the Polygamma functions ψ(n) (z) (n ∈ N), by using a known series representation of ψ(n) (z) in an elementary (yet technical) way. Ko¨ lbig [687, 688], in his CERN technical report, also gave two extensions of Gauss’s result to ψ(n) (z), by using the series definition of Polylogarithm function and the above-known series representation of ψ(n) (z). Recently, Choi and Cvijovic´ [271] presented a unified formula of ψ(n) (p/q) (p, q, n ∈ N; 1 5 p < q), which can be specialized in those formulas of Davis [368] and Ko¨ lbig [687, 688], expressed in terms of the Bernoulli polynomials given in 1.7 and the generalized Zeta functions in 2.2, which was shown in the following two ways:   (n) p ψ =(−1)n+1 n! qn q h i   q−1  n+1 X s 1+ 12 (n+1) (2π) · En (s; p ; q) (−1) Bn+1 2 · (n + 1)! q (57) s=0 #   q X 1 k + n+1 Fn (s; p ; q) En+1 (k; s ; q) ζ n + 1, q q k=1

(p, q, n ∈ N; 1 5 p < q), where [x] denotes the greatest integer 5 x, and     1 + (−1)n 2πsp 1 − (−1)n 2πsp En (s; p ; q) := sin + cos 2 q 2 q and Fn (s; p ; q) :=

1 + (−1)n cos 2



   2π sp 1 − (−1)n 2πsp + sin . q 2 q

Special Values of ψ(n) (z)   Since ψ(n) pq in (57) is expressed in terms of Bn (x) and ζ (s, a), in order to give special cases of (57) (see [688] and [9]), it is natural to know some of their properties. We demonstrate to give the value of the simple case of (57) when p = 1 and q = 2:     (n) 1 ψ = (−1)n+1 n! 2n+1 − 1 ζ (n + 1) (n ∈ N), (58) 2

Introduction and Preliminaries

35

which is obtained from the aid of the formulas 1.7(19), 2.3(3) and 2.3(18). It is noted that (58) is easily derived from (47) and 2.3(3). Likewise, we have (see [272])   1     √ 2n (2π)2n+1 1 3  n   = ± (−1) 3 3 B2n+1  2 2(2n + 1) 3   ψ(2n) 3  (2n)!  1 − 32n+1 ζ (2n + 1) (n ∈ N) + 2

(59)

  1     2n+1 1 4  n 2n (2π)   = ± (−1) 4 B2n+1  3 2n + 1 4   ψ(2n) 4   + (2n)! 1 − 22n+1 22n ζ (2n + 1)

(60)

ψ(2n)

and ψ(2n)

(n ∈ N).

We give the relations ψ(2n)

    1 1 = 2 ψ(2n) 6 3

ψ(2n)

      5 1 1 = −2 ψ(2n) − ψ(2n) 6 3 2

(n ∈ N)

(61)

and (n ∈ N),

(62)

which is easily obtained by recalling the multiplication formula (56) for ψ(n) (z). Finally, several further special values are: ψ(2n−1)

  1     4 

   = ± (2n − 1)! 2 3    ψ(2n−1)  4

4n−1

β(2n)

+ (−1)n−1 22n−2 (22n − 1)B2n

(63)

(2π)2n , 2n

where

β(s) :=

    ζ s, 14 − ζ s, 41 4s

.

(64)

36

Zeta and q-Zeta Functions and Associated Series and Integrals

The Asymptotic Expansion for ψ(z) From 1.1(34), we obtain the following asymptotic expansion for ψ(z): n

ψ(z) = log z −

  1 X B2k −2n−2 − + O z 2z 2k z2k

(65)

k=1

(|z| → ∞; | arg(z)| 5 π −  (0 <  < π); n ∈ N0 ). Now we shall show (see Barnes [94]) that n X

ψ(k) = n log n − n +

k=1

  1 + O n−1 2

(n → ∞)

(66)

(n → ∞),

(67)

and n X

  ψ 0 (k) = log n + 1 + γ + O n−1

k=1

where γ denotes the Euler-Mascheroni constant given in 1.1(3). Indeed, setting f (x) = 1/x and f (x) = 1/x2 with a = 1 in the Euler-Maclaurin summation formula 1.4??, we obtain 1+

1 1 1 1 15 + · · · + = γ + log n + − 2 + 4 − ··· 2 n 2n 8n 2n

(68)

1+

1 π2 1 1 1 1 1 + ··· + 2 = − + 2− 3+ − ··· , 2 6 n 2n 30 n5 2 n 6n

(69)

and

respectively. We can readily deduce from (69) that   1 1 1 1 −3 + + · · · = + + O n n 2 n2 n2 (n + 1)2

(n → ∞).

(70)

Since ψ(m) =

1 1 1 + + ··· + −γ 1 2 m−1

(m ∈ N \ {1}),

we can also write (66) in the form: n X k=1

ψ(k) = −nγ +

n−1 n−2 1 + + ··· + . 1 2 n−1

(71)

Introduction and Preliminaries

37

We, thus, obtain n X

  1 1 1 1 2 n−1 ψ(k) = n −γ + + + · · · + − − − ··· − 1 2 n−1 1 2 n−1 k=1 ! n−1 X 1 = n −γ + − n + 1, k k=1

which, in view of (68), yields n X

 ψ(k) = n −γ + γ + log(n − 1) +

k=1

  1 −2 +O n −n+1 2(n − 1)

(n → ∞),

which implies the result (66). In order to prove (67), we consider (54) with m = n = 1: ψ 0 (z + 1) = ψ 0 (z) −

1 , z2

(72)

the repeated application of which gives ψ 0 (m) = −



 1 1 1 + ψ 0 (1), + + · · · + 12 22 (m − 1)2

which may also follow directly from (54). Hence, we have n X

ψ 0 (k) = n ψ 0 (1) −

k=1

n−1 n−2 1 − 2 − ··· − . 12 2 (n − 1)2

(73)

Now apply the following special case of (53) when n = z = 1: ψ 0 (1) = ζ (2) =

π2 , 6

(74)

in conjunction with (68), and (73) readily yields n X

  π2 1 1 1 1 2 n−1 ψ (k) = n − 2 + 2 + ··· + + 2 + 2 + ··· + 6 1 2 (n − 1)2 1 2 (n − 1)2 k=1     1 1 −1 =n 2 + + · · · + γ + log(n − 1) + O n (n → ∞), n (n + 1)2 (75) 0

which, by applying (70), produces the desired result (67).

38

Zeta and q-Zeta Functions and Associated Series and Integrals

1.4 The Multiple Gamma Functions The double Gamma function 02 and the multiple Gamma functions 0n were defined and studied systematically by Barns [94–97] in about 1900. Before their investigation by Barnes, these functions had been introduced in a different form by (for example) Ho¨ lder [563], Alexeiewsky [18] and Kinkelin [666]. Although these functions did not appear in the tables of the most well-known special functions, the double Gamma function was cited in the exercises by Whittaker and Watson [1225, p. 264] and recorded also by Gradshteyn and Ryzhik [505, p. 661, Entry 6.441(4); p. 937, Entry 8.333]. In about the middle of the 1980s, these functions were revived in the study of the determinants of the Laplacians on the n–dimensional unit sphere Sn (see [260], [706], [881], [954], [1190], [1201]). Shintani [1026] also used the double Gamma function to prove the classical Kronecker limit formula. Friedman and Ruijsenaars [463] showed that Shintani’s work on multiple Zeta and Gamma functions can be simplified and extended by making use of difference equations. Its p-adic analytic extension appeared in a formula of Cassou-Nogue` s [221] for the p-adic L-functions at the point 0. Choi et al. (see [269, 291, 292]) used these functions to evaluate some families of series involving the Riemann Zeta function, as well as to compute the determinants of the Laplacians. Choi et al. [269] addressed the converse problem and applied various formulas for series associated with the Zeta and related functions with a view to developing the corresponding theory of multiple Gamma functions. Adamchik [8] discussed some theoretical aspects of the multiple Gamma functions and their applications to summation of series and infinite products. Matsumoto [804] proved several asymptotic expansions of the Barnes double Zeta function and the double Gamma function and presented an application to the Hecke L-functions of real quadratic fields. Ruijsenaars [990] showed how various known results concerning the Barnes multiple Zeta and Gamma functions can be obtained as specializations of the simple features shared by a quite remarkably extensive class of functions.

The Double Gamma Function 02 Here, we summarize some properties of the double Gamma and related functions. We also introduce some mathematical constants associated with the double and triple Gamma functions. Barnes [94] defined the double Gamma function 02 = 1/G, satisfying each of the following properties: (a) G(z + 1) = 0(z)G(z) (z ∈ C); (b) G(1) = 1; (c) Asymptotically,

log G(z + n + 2) =

 2  n+1+z n 5 z2 log(2π) + +n+ + + (n + 1)z log n 2 2 12 2   2 3n 1 − − n − nz − log A + + O n−1 (n → ∞), 4 12

(1)

Introduction and Preliminaries

39

where 0 is the familiar Gamma function introduced in Section 1.1 and A is called the Glaisher-Kinkelin constant, defined by

log A = lim

n→∞

( n X k=1

)  1 n2 n2 n , + + log n + k log k − 2 2 12 4 

(2)

the numerical value of A being given by A∼ = 1.282427130 · · · . From this definition, Barnes [94] deduced several explicit Weierstrass canonical product forms of the double Gamma function 02 , one of which is recalled here in the form: {02 (z + 1)}−1 = G(z + 1) Y    ∞  1 z k 1 1 z2 = (2π) 2 z exp − z − (γ + 1)z2 1+ exp −z + , 2 2 k 2k

(3)

k=1

where γ denotes the Euler-Mascheroni constant given by 1.1(3). Barnes [94] also gave the following two more equivalent forms of the double Gamma function 02 : {02 (z + 1)}−1 = G(z + 1) Y    ∞ 1 0(k) 1 1 1 = (2π) 2 z exp − z(z + 1) − γ z2 exp zψ(k) + z2 ψ0 (k) ; 2 2 0(z + k) 2 k=1

(4)   2   2 1 z π γ− +1+γ z− 0(z + 1) 2 6 2    ∞ Y ∞ 0 Y z2 z z + · 1+ exp − , m+n m + n 2(m + n)2 1

{02 (z + 1)}−1 = G(z + 1) = (2π) 2 z exp

m=0 n=0

(5) where the prime denotes the exclusion of the case n = m = 0 and the Psi (or Digamma) function ψ is given by 1.2(1). Each form of these products is convergent for all finite values of |z|, by the Weierstrass factorization theorem (see Conway [339, p. 170]). The double Gamma function satisfies the following relations: G(1) = 1

and G(z + 1) = 0(z)G(z) (z ∈ C).

(6)

40

Zeta and q-Zeta Functions and Associated Series and Integrals

For sufficiently large real x and a ∈ C, we have the Stirling formula for the G-function: log G(x + a + 1) =

x+a 1 3x2 log(2π) − log A + − − ax 2 12 4   2   1 a2 x − + + ax log x + O x−1 (x → ∞). + 2 12 2

(7)

The following special values of G (see Barnes [94]) may be recalled here: G

  1 3 1 1 1 = 2 24 · π − 4 · e 8 · A− 2 ; 2

G(n + 2) = 1! 2! · · · n!

and G(n + 1) =

(8) (n!)n 1 · 2 · 32 · 43 · · · nn−1

(n ∈ N).

(9)

We shall deduce only the expression (4) (see Barnes [94]). Indeed, taking the logarithmic derivative on both sides of the fundamental functional relation in (6), with respect to z, we obtain G0 (z + 1) 0 0 (z) G0 (z) = + , G(z + 1) 0(z) G(z) from which n

G0 (z + n + 2) X 0 0 (z + k + 1) G0 (z + 1) = + . G(z + n + 2) 0(z + k + 1) G(z + 1) k=0

For sufficiently small values of |z|, by Taylor’s theorem, we have d 0 0 (1 + k) z2 d2 0 0 (1 + k) 0 0 (z + 1 + k) 0 0 (1 + k) = +z + + ··· , 0(z + 1 + k) 0(1 + k) dk 0(1 + k) 2! dk2 0(1 + k) provided that the coefficients in the expansion are finite. Conversely, it follows from 1.1(2) or 1.3(3) that ∞

−

X 0 0 (1 + z) =γ + 0(1 + z)

m=1



 1 1 − , z+m m

which, upon differentiating r times with respect to z and setting z = k in the resulting equation, yields ∞

X dr 0 0 (1 + k) 1 = (−1)r−1 r! r dk 0(1 + k) (m + k)r+1 m=1

(r ∈ N\{1}),

Introduction and Preliminaries

41

so that d 0 0 (1 + k) 0 0 (z + 1 + k) 0 0 (1 + k) − −z 0(z + 1 + k) 0(1 + k) dk 0(1 + k) ( ) ∞ ∞ X X 1 (−1)r−1 = zr , (m + k)r+1 m=1

r=2

from which we find that n  0 X 0 (z + 1 + k)

0 0 (1 + k) d 0 0 (1 + k) − −z 0(z + 1 + k) 0(1 + k) dk 0(1 + k) k=0 ) ( ∞ n X ∞ X X 1 r r−1 z , = (−1) (m + k)r+1 r=2



k=0 m=1

where we are not rearranging a double series, if n is not actually infinite. Now it is not difficult to see from Eisenstein’s theorem (see Forsyth [457, p. 87]) that lim

n→∞

n X ∞ X k=0 m=1

1 (m + k)r+1

is convergent when r = 2. Hence, lim

n→∞

n  0 X 0 (z + 1 + k) k=0

0 0 (1 + k) d 0 0 (1 + k) − −z 0(z + 1 + k) 0(1 + k) dk 0(1 + k)



is finite, when |z| < 1 (z ∈ C). We also have the following relation:  n  d 0 0 (1 + k) G0 (z + 1) X 0 0 (z + 1 + k) 0 0 (1 + k) = − −z − G(z + 1) 0(z + 1 + k) 0(1 + k) dk 0(1 + k) k=0   n G0 (z + n + 2) X 0 0 (1 + k) d 0 0 (1 + k) − + +z . G(z + n + 2) 0(1 + k) dk 0(1 + k) k=0

Therefore, for sufficiently small values of |z|, we may take G0 (z + 1) = α + 2βz G(z + 1) "∞   # (10) Y 0(z + 1 + k) d 0 0 (1 + k) z2 d 0 0 (1 + k) + log exp −z − , dz 0(1 + k) 0(1 + k) 2 dk 0(1 + k) −

k=0

42

Zeta and q-Zeta Functions and Associated Series and Integrals

provided that

− lim

n→∞

 n  0 X 0 (1 + k) d 0 0 (1 + k) G0 (z + n + 2) = −α − 2βz + lim +z , n→∞ G(z + n + 2) 0(1 + k) dk 0(1 + k) k=0

and provided also that the expression on the right-hand side tends to the same value as that obtainable from (1). Thus, we have   G0 (z + n + 2) 1 = (n + 1 + z) log n + log(2π) − n + O n−1 G(z + n + 2) 2

(n → ∞).

If, then, we can prove that, for suitable values of α and β, n  0 X 0 (1 + k)

 d 0 0 (1 + k) − α − 2βz + +z 0(1 + k) dk 0(1 + k) k=0   1 = (n + 1 + z) log n + log(2π) − n + O n−1 (n → ∞), 2

(11)

we shall have shown that, for suitable values of α and β, (10) holds true from the fact (see Barnes [94, p. 269]) that G0 (z + 1)/G(z + 1) is the only solution of f (z + 1) = f (z) +

0 0 (z) , 0(z)

which has such an asymptotic expansion near at infinity. Now the left-hand side of (11) may be written as

−α − 2βz +

n+1 X  ψ(k) + z ψ 0 (k) , k=1

which, in view of 1.3(57) and 1.3(58), reduces at once to (n + 1) log n − n + z (log n + 1 + γ − 2β) +

  1 − α + O n−1 2

(n → ∞),

which, upon putting α=

1 1 1+γ − log (2π) and β = , 2 2 2

becomes the right-hand side of (7).

(12)

Introduction and Preliminaries

43

If we now substitute the values of α and β from (12) into (10), we obtain the following expression: G0 (z + 1) 1 1 = − log(2π) + (1 + γ ) z G(z + 1) 2 2 "∞   # Y 0(z + 1 + k) d 1 2 0 + log exp −z ψ(1 + k) − z ψ (1 + k) dz 0(1 + k) 2 −

k=0

(|z| < 1; z ∈ C), which, upon integrating with respect to z and obtaining the value of the constant of integration by making z = 0, finally yields the desired expression (4) valid for |z| < 1. In fact, (4) holds true for all finite values of |z| by the principle of analytic continuation. From (4), we may also evaluate a series involving the ψ-function: ∞  X

ψ(k) +

k=1

 1 0 γ 1 ψ (k) − log k = 1 + − log(2π) 2 2 2

(13)

by taking logarithms on both sides of (4) and setting z = 1 in the resulting equation. In view of 1.2(53), the left-hand side of (13) can be written in its equivalent form: ∞  X

ψ(k) +

k=1

 1 γ 1 ζ (2, k) − log k = 1 + − log(2π). 2 2 2

(14)

In view of their need in evaluating integrals involving the double Gamma function, it may be important to know other special values of the G-function. Here, we give two special values of the G-function as an illustration. The following special value of the 0-function is known (see Spiegel [1058, p. 1]):   1 ∼ 0 (15) = 3.62560 99082 21908 · · · . 4 The Catalan constant G is defined by 1 G := 2

Z1 K(k)dk =

∞ X (−1)m ∼ = 0.91596 55941 77219 015 · · · , (2m + 1)2

(16)

m=0

0

where K is the complete elliptic integral of the first kind, given by (cf. Equation 1.5(33)) Zπ/2 K(k) := 0

dt √ 1 − k2 sin 2 t

(|k| < 1).

(17)

44

Zeta and q-Zeta Functions and Associated Series and Integrals

The following integral is known (see Gradshteyn and Ryzhik [505, p. 526, Entry 4.224]): Zπ/4 1 π log sin t dt = − log 2 − G. 4 2

(18)

0

Considering (6), (18) and (28), we obtain       1 1 G 1 1 3 = 2− 8 · π − 4 · e 2π · 0 G . G 4 4 4

(19)

We also recall a duplication formula for the G-function (see Choi [263, p. 290]):    1 1 11 1 2 2 G(z) G z + G(z + 1) = e 4 · A−3 · 2−2z +3z− 12 · π z− 2 · G(2z), 2

(20)

which is an obvious special case (n = 2) of the following multiplication formula given by Barnes [94, p. 291]: n−1 Y n−1 Y i=0 j=0

  1 2 2 1 i+j = K(n) (2π) 2 n(n−1)z nnz− 2 n z G(nz), G z+ n

(21)

where, for convenience, K(n) := A1−n e 12 (n 2

Setting z = 0

1 4

1

2 −1)

(2π)− 2 (n−1) n− 12 . 1

5

in (20), and using (6) and (8), we obtain

      2 1 1 3 9 1 3 1 G G = 2− 4 ·π − 2 · e 8 · A− 2 . 4 4 4

(22)

Combining (19) and (22), we obtain     − 3 4 3 G 1 1 − 4π − 98 ∼ 32 =e ·A 0 G = 0.293756 · · · 4 4

(23)

or, equivalently, G

   1   4 1 1 3 G 9 1 3 ∼ = 2− 8 · π − 4 ·e 32 + π · A− 8 0 = 0.848718 · · · . 4 4

It follows from (23), (24) and (6) that   G 34 1 1 G
= 2− 8 · π − 4 · e 2π . G 54

(24)

(25)

Introduction and Preliminaries

45

Integral Formulas Involving the Double Gamma Function We begin by recalling the following integral formula: Zz

πt cot π t dt = z log(2π) + log

0

G(1 − z) , G(1 + z)

(26)

which is due, originally, to Kinkelin [666]. Indeed, in view of (3), if we set ) (  ∞  Y 1 + kz k −2z G(1 + z) z −z e = (2π) e 8(z) := G(1 − z) 1 − kz k=1

and differentiate logarithmically with respect to z, and apply the known expansion (see Ahlfors [13, p. 188]): πz cot πz = 1 + 2

∞ X n=1

z2 (z 6∈ Z), z2 − n2

(27)

we shall readily obtain d log 8(z) = log(2π) − π z cot π z (z 6∈ Z), dz which, upon integration, yields (26), since 8(0) = 1. Some simple consequences of Kinkelin’s formula (26) are worthy of note here (see also Barnes [94, p. 279]). First, by using integration by parts in (26), we have Zz 0

 sin πz G(1 + z) log sin πt dt = z log + log , 2π G(1 − z)

which, upon setting t = Zz 0



1 2

− u and replacing z by

1 2

(28)

− z, yields

  1       G + z 2 cos π z 1 1 1 . log − log 2 − log 0 − z + log  log cos π t dt = z − 2 2π 2 2 G 1 −z 2

(29) Making use of (29), we obtain the following analogue of (26): Zz 0

  1     G + z 2 1 cos πz 1 , π t tan πt dt = − log − z log(2π) − log 0 − z + log  2 π 2 G 1 −z 2

(30)

46

Zeta and q-Zeta Functions and Associated Series and Integrals

which would follow also from (26) by setting t = 12 − u and replacing z by Combining (28) and (29), we readily have the integral formula: Zz

1 2

− z.

  1 cos π z 1 log + log 0 −z 2 π 2   G 12 + z G (1 + z) . + log − log  G (1 − z) G 1 −z

log tan πt dt = z log tan πz +

0

(31)

2

Similarly, by using various trigonometric identities, we can obtain the following integral formulas: Zz  0

Zz  0

Zz

πt cos π t

πt sin π t

2

2

cos π z + 2z log(2π) π   1   G + z 2 1 ; + 2 log 0 − z − 2 log  2 G 12 − z

dt = π z2 tan π z + log

dt = −π z2 cot πz + 2z log(2π) − 2 log

πt tan π t dt = log

0

G (1 − z) . G (1 + z)

G (1 − z) 1 G (1 − 2z) − log , G (1 + z) 2 G (1 + 2z)

(32)

(33)

(34)

which, in view of the known duplication formula (20) for the G-function, is the same as (31); Zz 0

Zz 0

Zz 0

  G 1 − 12 z G (1 − z) πt  − log dt = z log(2π) + 4 log  ; sin πt G (1 + z) G 1+ 1 z

(35)

2

π 2 z3 cos π z + π z2 tan πz + log 3 π   G 32 − z ; + 2z log(2π) + 2 log  G 21 + z

(πt tan π t)2 dt = −

(πt cot π t)2 dt = −

π 2 z3 G (1 − z) ; − π z2 cot π z + 2z log(2π) + 2 log 3 G (1 + z)

(36)

(37)

Introduction and Preliminaries

Zz  0

πt sin π t

2

cos πt dt = 2

47

Zz/2 0

πt sin πt

Zz/2

2 dt − 2

0

πt cos πt

2 dt

 
G 12 + 2z cos π2z G 1 − 2z πz2
+ 4 log  ; =− − 2 log + 4 log sin πz π G 1 + 2z G 32 − 2z   1 Zz G + z 2 πt 1 cos π z G(1 − z) . dt = − log + log + log  sin πt cos π t 2 π G(1 + z) G 3 −z

(38)

(39)

2

0

Replacing t by at/π in (28), we obtain Zz 0


G 1 + πa z sin az π
, log sin at dt = z log + log 2π a G 1 − πa z

(40)

which, in view of the trigonometric identity:     1 1 (a + b)t sin (a − b)t , cos bt − cos at = 2 sin 2 2 yields Zz

sin log(cos bt − cos at) dt = z log

0

h

i

1 2 (a + b)z

sin

h

1 2 (a − b)z

i

2π 2     G 1 + a+b G 1 + a−b 2π z 2π z 2π 2π + . + log  log  a−b a+b a−b G 1 − a+b z G 1 − z 2π 2π

(41)

By differentiating Alexeiewsky’s theorem (see Barnes [94, p. 281]): Zz

z2 1 log 0(t + 1) dt = [log(2π) − 1]z − + z log 0(z + 1) − log G(z + 1) 2 2

(42)

0

with respect to z, we obtain d 1 log G(z) = [log(2π) + 1] − z + (z − 1)ψ(z), dz 2

(43)

and so we have ∂ log G(h(az)) ∂a   1 ∂ = [log(2π) + 1] − h(az) + [h(az) − 1]ψ(h(az)) h(az), 2 ∂a

(44)

48

Zeta and q-Zeta Functions and Associated Series and Integrals

where h(z) is a function of z and a is a constant. Differentiating each side of (41) with respect to a and using (44) and 1.3(9), we obtain Zz 0

2az tsin at dt = 2 log(2π) cos bt − cos at a − b2     a+b a−b G 1 − G 1 − z z 2π 2π 2π 2π − . log  log  − 2 a+b a−b (a + b)2 (a − b) z z G 1+ G 1+ 2π

Replacing t by Zz 0

1 π

(45)

2π

arctan at in (31), we obtain

    log t π log z cos arctan az dt = arctan az + log a 2a π 1 + a2 t2     1 3 1 G 1 + arctan az G − arctan az π 2 π π   . + log  1 1 a G 1 − arctan az G + 1 arctan az π

2

(46)

π

Setting a = 1 and z = 1 in (46) and using 1.3(4), we obtain Z1 0

log t dt = −G. 1 + t2

(47)

It is easy to verify that Zz 0

  a 2a 1−a 1 dt = arctan tan z 1+a 2 1 + a2 + 2a cos t 1 − a2

and Zz 0

  cos t z 1 + a2 1−a 1 dt = − arctan tan z . 2a a(1 − a2 ) 1+a 2 1 + a2 + 2a cos t

Using these identities and applying the Leibniz rule (see De Lillo [372, p. 665]) to the following integral and again integrating the resulting equation with respect to a

Introduction and Preliminaries

49

from 1 to a, we obtain Zz 0

   Za 1 dt 1−t 1 + a2 + 2a cos t dt = z log a − 2 arctan tan z log 2 + 2 cos t 1+t 2 t 

1

= (2z − π) log a + 2

Za



t + cos z sin z

arctan



dt . t

1

Equivalently, from (29) and the following integral: Zz

 log(2 + 2 cos t) dt = (2z − 2π) log 

cos



1 2z

π

0

  + 4π log

G



1 2

z + 2π

G



3 2

z − 2π

1 2



 ,

we have Zz

 log(1 + a2 + 2a cos t) dt = π loga + (2z − 2π) log 

a cos



1 2

z + 2π



Za

 +2 + 4πlog  z G 32 − 2π 1

z



π

0

G





t + cos z arctan sin z



dt . t

Applying integration by parts to the last integral gives us the following equivalent form: Zz

 log(1 + a2 + 2a cos t) dt = π log a + (z − π) log 

a2 cot



1 2

z

 

2π 2

0

  z    G 12 + 2π a + cos z a  + 2 arctan log + 4π log  sin z sin z G 3− z 

2

a+cos z sin z

Z

−2 cot

1 2z

log(t − cot z) dt. 1 + t2

2π

(48)

50

Zeta and q-Zeta Functions and Associated Series and Integrals

Setting z = 21 π in (48) and using 1.3(4), (46) and (47), we obtain   Zπ/2 cos (arctan a) 2 log(1 + a + 2a cos t) dt = −π log π 0     G 1 + π1 arctan a G 32 − π1 arctan a   . − 2π log  G 1 − π1 arctan a G 12 + π1 arctan a

(49)

Integrating by parts and using (46), we have Zz

  arctan at π cos (arctan az) dt = − log t 2 π 0     G 1 + π1 arctan az G 32 − π1 arctan az   . − π log  G 1 − π1 arctan az G 12 + π1 arctan az

Replacing t by

Zz 0

1 π

(50)

arcsin at in (28), we get

log t dt = √ 1 − a2 t2



arcsin az a



   z  π G 1 + π1 arcsin az . log + log  2π a G 1 − π1 arcsin az

(51)

Using (51) and the following relation: Zz

arcsin at dt = (arcsin az) log z − a t

Zz 0

0

log t dt , √ 1 − a2 t2

we obtain Zz 0

  G 1 − π1 arcsin az arcsin at . dt = log(2π)(arcsin az) + π log  t G 1 + π1 arcsin az

(52)

Introduction and Preliminaries

51

Integrating by parts and using (46), we find that Zz

(arctan at)2 dt t2

0

    (arctan az)2 cos (arctan az) − π a log − a log 1 + a2 z2 arctan az z π     G 1 − π1 arctan az G 12 + π1 arctan az    + 2π a log  G 1 + π1 arctan az G 32 − π1 arctan az

=−

2

Zz

+a

0

log(1 + a2 t2 ) dt. 1 + a2 t2

(53)

Numerous special cases of the integral formulas (considered in this section) include the following results: Zπ/2

x 2 π2 cos x dx = − + 4G; sin x 4

(54)

0

Z1/2 1 7 1 3 log 0(t + 1)dt = − − log 2 + log π + log A; 2 24 4 2

(55)

0

Z1/4 1 9 G 1 3 log 0(t + 1)dt = − − log 2 + log π + log A + ; 4 8 8 8 π

(56)

0

Z1

arctan x dx = G; x

(57)

0 √ 1/ Z 2

π G arcsin x dx = log 2 + ; x 8 2

(58)

0

Zπ/4 π G x cot x dx = log 2 + ; 8 2

(59)

0

Zπ/4 π G log cos x dx = − log 2 + ; 4 2 0

(60)

52

Zeta and q-Zeta Functions and Associated Series and Integrals

Zπ/4 G π x tan x dx = − log 2 + ; 8 2

(61)

0

Zπ/4 log tan x dx = −G;

(62)

0

Zπ/4

x 2 π2 π dx = + log 2 − G; cos x 16 4

(63)

x 2 π2 π dx = − + log 2 + G; sin x 16 4

(64)

Zπ/4 π3 π2 π x2 tan2 x dx = − + + log 2 − G; 192 16 4

(65)

0

Zπ/4 0

0

Zπ/4 π3 π2 π x2 cot 2 x dx = − − + log 2 + G; 192 16 4

(66)

0

Zπ/2

x dx = 2G. sin x

(67)

0

The Evaluation of an Integral Involving log G(z) First, we shall introduce two interesting mathematical constants, in addition to the Glaisher-Kinkelin constant A, by recalling the Euler-Maclaurin summation formula (cf. Hardy [537, p. 318]): n X k=1

f (k) ∼ C0 +

Zn a

∞

f (x) dx +

X B2r 1 f (n) + f (2r−1) (n), 2 (2r)!

(68)

r=1

where C0 is an arbitrary constant to be determined in each special case and 1 1 1 1 1 5 B0 = 1, B1 = − , B2 = , B4 = − , B6 = , B8 = − , B10 = , · · · , 2 6 30 42 30 66 and B2n+1 = 0

(n ∈ N)

are the Bernoulli numbers (see Section 1.7). For another useful summation formula, see Edwards [399, p. 117].

Introduction and Preliminaries

53

Letting f (x) = x2 log x and f (x) = x3 log x in (68) with a = 1, we obtain " log B = lim

n→∞

n X

#  n3 n2 n n3 n k log k − + + log n + − 3 2 6 9 12

(69)

#  4 3 2 4 2 n n n 1 n n k3 log k − + + − log n + − , 4 2 4 120 16 12

(70)

2

k=1



and " log C = lim

n→∞

n X k=1



respectively; here, B and C are constants, whose approximate numerical values are given by B∼ = 1.03091 675 · · ·

(71)

C∼ = 0.97955 746 · · · .

(72)

and

The constants B and C were considered, recently, by Choi and Srivastava [289, 292]. See also Adamchik [6, p. 199]. Bendersky [114] presented a set of constants, including B and C. It is known that (cf. Barnes [94, p. 288]) Z1 log G(t + 1)dt =

1 1 + log(2π) − 2 log A. 12 4

(73)

0

Setting z = t in the logarithmic form of (3) and integrating both sides of the resulting equation from t = 0 to t = 12 , we obtain 1

Z2 log G(t + 1)dt =

1 γ 1 log(2π) − − 16 48 12

0

 ∞     1X  2 1 1 2 + 4k + 2k log(2k + 1) − 4k + 2k log(2k) − 2k − + . 4 2 12k k=1

(74)

54

Zeta and q-Zeta Functions and Associated Series and Integrals

Consider the sum Sn given by

Sn :=

n h    i X 4k2 + 2k log(2k + 1) − 4k2 + 2k log(2k) k=1

n h i X = (2k + 1)2 log(2k + 1) − (2k + 1) log(2k + 1) − (2k)2 log(2k) − 2k log(2k) k=1

=

=

"2n+1 X

n 2n+1 X X k log k − 2 (2k)2 log(2k) − k log k

#

2

k=1

k=1

"2n+1 X

n X

k2 log k − 8

k=1

k=1

k2 log k −

k=1

2n+1 X k=1

k log k − 8 log 2

n X

# k2 ,

k=1

which, in view of (2) and (69), yields    (2n + 1)3 (2n + 1)2 (2n + 1) + + log(2n + 1) Sn = log B + 3 2 6   (2n + 1)3 (2n + 1) n3 n2 n 8 8 − + − 8 log B − 8 + + log n + n3 − n 9 12 3 2 6 9 12 (75)  (2n + 1)2 (2n + 1) 1 + + log(2n + 1) 2 2 12    n(n + 1)(2n + 1) (2n + 1)2 − 8 log 2 + + O n−1 (n → ∞) 4 6   2 4 1 8 3 = − log A − 7 log B + + lim n + 4n2 + n − log(2n + 1) 9 n→∞ 3 3 12   8 3 4 1 1 − n + 4n2 + n log n − n2 − n 3 3 3 6        4 8 log 2 n3 − (4 log 2)n2 − log 2 n + O n−1 (n → ∞). − 3 3 

− log A −

Now, substituting from (75) and recalling 1.1(3) and applying the following expansion:   1 1 1 1 dn − (0 < dn < 1; n ∈ N), log 1 + = − 2+ 3 2n 2n 8n 24n 64n4

(76)

Introduction and Preliminaries

55

we evaluate the summation part of (74) as follows: 2 Tn := − log A − 7 log B + 9   8 3 4 1 2 + n + 4n + n − log(2n + 1) 3 3 12   4 1 1 8 3 n + 4n2 + n log n − n2 − n − 3 3 3 6     8 4 3 2 − log 2 n − (4 log 2)n − log 2 n 3 3    n(n + 1) n 1 −2 − + (γ + log n) + O n−1 (n → ∞) 2 2 12 1 γ = − log A − 7 log B − 12 12  4 1 8 3 n + 4n2 + n − log 2 + 3 3 12     4 1 1 8 3 + n + 4n2 + n − log 1 + 3 3 12 2n          4 5 8 4 3 2 − log 2 n − 4 log 2 + n − log 2 + n + O n−1 (n → ∞). 3 3 3 3 Thus, we have Tn =

γ 1 − log A − 7 log B − 12 12  4 1 4 5 5 8 3 + n + 4n2 + n − log 2 + n2 + n + 3 3 12 3 3 18          8 4 4 5 3 2 − log 2 n − 4 log 2 + n − log 2 + n + O n−1 (n → ∞), 3 3 3 3

which, upon taking the limit as n → ∞ and using (74), yields our desired result: 1

Z2 log G(t + 1)dt =

1 1 7 1 (log 2 + 1) + log π − log A − log B, 24 16 4 4

(77)

0

in terms of the mathematical constant B defined by (69). Vigne´ ras [1199] gave an integral representation for the double Gamma function 02 : log 02 (z + 1) = −

Z∞ 0

  z2 t 2 −zt 1 − zt − − e dt
2 2 t 1 − e−t e−t

+ (1 + γ )

z2 2

−

3 log π 2

(<(z) > −1).

(78)

56

Zeta and q-Zeta Functions and Associated Series and Integrals

The Multiple Gamma Functions There are two known ways to define an n-ple Gamma functions 0n : Barnes [97] (see also Vardi [1190]) defined 0n by using the n-ple Hurwitz Zeta functions (see Choi and Quine [278]; also Seo et al. [1019]); a recurrence formula of the Weierstrass canonical product forms of the n-ple Gamma functions 0n was given by Vigne´ ras [1199], who used the theorem of Dufresnoy and Pisot [395], which provides the existence, uniqueness and expansion of the series of Weierstrass satisfying a certain functional equation. By making use of the aforementioned Dufresnoy-Pisot theorem and starting with f1 (x) = −γ x +

∞ n X x

 x o − log 1 + , n n

n=1

(79)

Vigne´ ras [1199] obtained a recurrence formula of 0n (n ∈ N), which is given by Theorem 1.4 The n-ple Gamma functions 0n are defined by 0n (z) = {Gn (z)}(−1)

n−1

(n ∈ N),

(80)

where Gn (z + 1) = exp ( fn (z))

(81)

and the functions fn (z) are given by fn (z) = −z An (1) +

n−1 i X pk (z) h (k) fn−1 (0) − A(k) n (1) + An (z), k!

(82)

k=1

with "  n  n−1 1 z 1 z An (z) = − + ··· n L(m) n − 1 L(m) m∈N0 n−1 ×N   z n n−1 z + (−1) log 1 + + (−1) , L(m) L(m) X

where L(m) = m1 + m2 + · · · + mn , if m = (m1 , m2 , . . . , mn ) ∈ N0 n−1 × N

(83)

Introduction and Preliminaries

57

and the polynomials pn (z) given by

pn (z) :=

 n 1 + 2n + 3n + · · · + (N − 1)n  

(z = N; N ∈ N \ {1})

  Bn+1 (z) − Bn+1 n+1

(z ∈ C)

(84)

satisfy the following relations: p0n (z) =

B0n+1 (z) n+1

= Bn (z) and

pn (0) = 0,

Bn (z) being the Bernoulli polynomial of degree n in z. Clearly, we find from the definition (84) that  n+1  1 X n+1 pn (z) = Bn+1−k zk n+1 k

(n ∈ N).

(85)

k=1

By analogy with the Bohr-Mollerup (Theorem 1.1), which guarantees the uniqueness of the Gamma function 0, one can give for the double Gamma function and (more generally) for the multiple Gamma functions of order n (n ∈ N) a definition of Artin by means of the following theorem (see Vigne´ ras [1199, p. 239]): Theorem 1.5 For all n ∈ N, there exists a unique meromorphic function Gn (z) satisfying each of the following properties: (a) Gn (z + 1) = Gn−1 (z) Gn (z) (z ∈ C); (b) Gn (1) = 1; (c) For x = 1, Gn (x) are infinitely differentiable and dn+1 {log Gn (x)} = 0; dxn+1 (d) G0 (x) = x.

It may be remarked in passing that G1 (z) = 01 (z) = 0(z) and the case n = 2 of (80) can readily be seen to produce the double Gamma function (5). We observe that the function {0n (z)}−1 defined in (80) is an entire function of order n, all of whose zeros are the nonpositive integers 0, −1, −2, . . . , just as the Gamma function {0(z)}−1 , and the multiplicity of the zeros of {0n (z)}−1 at z = −k (k ∈ N0 ) is equal to the number of solutions of the equation:   L(m) = k + 1 m ∈ N0 n−1 × N .

58

Zeta and q-Zeta Functions and Associated Series and Integrals

The number of the solutions for m ∈ N0 n is the coefficient of xk+1 in the Maclaurin series expansion of   n+k (1 − x)−n , that is, . k+1 Therefore, the number of the solutions for m ∈ N0 n−1 × N is equal to       n+k n+k−1 n+k−1 − = . k+1 k+1 n−1 We, thus, conclude that {0n (z)}−1 is an entire function with zeros at z = −k (k ∈ N0 ), whose multiplicities are   n+k−1 (n ∈ N; k ∈ N0 ). (86) n−1

The Triple Gamma Function 03 When n = 3 in (80), we readily obtain an explicit form of the triple Gamma function 03 : 03 (1 + z) =G3 (1 + z)       π2 3 3 1 1 2 1 γ+ + z + γ + log(2π) + z + z = exp − 6 6 2 4 2 ( "   2  3 #) −1 Y z z 1 z 1 z exp − + , 1+ · L(m) L(m) 2 L(m) 3 L(m) 2 m∈N0 ×N

(87) where =

1 12



3 π2 − γ − 3 log(2π) + 2 12

∞

 +

1X ζ (n + 2) (−1)n 2 (n + 3)(n + 4) n=1

and L(m) = m1 + m2 + m3

with m = (m1 , m2 , m3 ) ∈ N0 2 × N,

ζ (s) being the Riemann zeta function (see Section 2.3). Now the infinite sum in  can be evaluated explicitly by using a known formula (see Choi and Srivastava [288, p. 116, Eq. (2.63)]): ∞ X (−1)k k=3

ζ (k) 1 γ π2 = + − − 2 log A. (k + 1)(k + 2) 2 6 72

(88)

Introduction and Preliminaries

59

We, thus, find that =

3 1 − log(2π) − log A 8 4

(89)

in terms of the Glaisher-Kinkelin constant A defined by (2). If we set n = 3 in (86), we observe that {03 (z)}−1 is an entire function with zeros at z = −k (k ∈ N0 ), whose multiplicity is  1 2 k + 3k + 2 2

(k ∈ N0 ).

(90)

Furthermore, in view of (90), (87) can be written in the following equivalent form analogous to (3): 03 (1 + z) = G3 (1 + z) ∞  Y z − 12 k(k+1) R(z) =e 1+ k k=1       1 1 2 1 1 3 1 , · exp (k + 1)z − 1+ z + 1+ z 2 4 k 6k k

(91)

where, for convenience, !     1 π2 3 3 1 1 2 3 1 R(z) := − γ+ + z + γ + log(2π ) + z + − log(2π) − log A z. 6 6 2 4 2 8 4

It follows that 03 satisfies several basic properties and characteristics, which are summarized here in Theorem 1.6 The triple Gamma function 03 is the unique meromorphic function satisfying each of the following properties: (a) 03 (1) = 1; (b) 03 (z + 1) = G(z)03 (z) (z ∈ C); (c) For x = 1, 03 (x) is infinitely differentiable and d4 {log 03 (x)} = 0. dx4

  Just as in (8), we now proceed to express the value of 03 21 in terms of the mathematical constants π, e, A and B. We begin by recalling the following known results: " n #   X 1 1 log(2π) = lim log k − n + log n + n n→∞ 2 2 k=1

(92)

60

Zeta and q-Zeta Functions and Associated Series and Integrals

and     1 1 1 1 = − 2 + 3 + O n−4 (n → ∞). log 1 + n n 2n 3n By taking logarithms on both sides of (91) and setting z = tion, if we make use of 1.1(3), we obtain

(93)

1 2

in the resulting equa-

 3 1 1 1 = log 03 1 + − log(2π) − log A 2 16 16 2 " n #   X k(k + 1) 1 n2 5 1 + lim − log 1 + + + n− log n . n→∞ 2 2k 8 16 24 

k=1

We, first, consider the following sum:

Un : =

n X k(k + 1) k=1 n X

1 2

=

2

  1 log 1 + 2k

k(k + 1) log(2k + 1)

k=1 n

n

n

log 2 X 1X 2 1X k(k + 1) − k log k − k log k 2 2 2 k=1 k=1 k=1 ! n n X 1 X 2 = (2k + 1) log(2k + 1) − log(2k + 1) 8 −

k=1

−

k=1

n log 2 X

2

k(k + 1) −

k=1

n 1X

2

n

k2 log k −

k=1

1X k log k 2 k=1

and so

Un =

1 8 −

2n+1 X

k2 log k − 4

k=1 2n+1 X

k2 log k − 4 log 2

k=1

log k +

k=1

−

n X

n X

n X

k2

k=1

! log k + n log 2

k=1 n

n

n

k=1

k=1

k=1

1X 2 1X log 2 X k(k + 1) − k log k − k log k, 2 2 2

(94)

Introduction and Preliminaries

61

which immediately leads us to Un =

2n+1 2n+1 n n X 1X 1X 1X 2 k log k − log k − k2 log k − k log k 8 8 2 k=1 k=1 k=1 k=1   3 n 1X 3 2 7 n + + n + n log 2. log k − 8 3 4 24

(95)

k=1

Upon substituting from (95) into (94), if we apply (2), (69) and (92), we obtain   3 1 7 1 = − log(2π) + log B log 03 1 + 2 16 16 8   3    n 3 7 1 1 + lim − + n2 + n − log 1 + n→∞ 3 4 24 16 2n  2 n n 35 1 + + − + log 2 6 3 288 16 1 7 3 − log π + log B = 16 16 8  2    n2 n n n 19 35 + lim − − − + O n−1 + + − n→∞ 6 3 288 6 3 288 7 1 = − log π + log B, 16 8 where we have also used (93) for the second equality. Thus, we find that   1 7 1 03 1 + = π − 16 · B 8 , 2 which, in view of (8) and the assertion (b) of Theorem 1.5, yields   1 3 1 3 7 1 03 = 2− 24 · π 16 · e− 8 · A 2 · B 8 . 2

(96)

(97)

A Multiplication Formula for the 0n Nishizawa [867] obtained a multiplication formula for the n-ple Gamma function 0n , by using his product formula for the multiple Gamma function 0n and other asymptotic formulas. Here, by employing the same method used by Choi and Quine [278], Choi and Srivastava [300] showed how the following multiplication formula for the multiple Gamma function 0n can be obtained rather easily and nicely: fn (z, p, r) 0n (z) = ζ (0,z) pn

p−1 Y q1 ,··· ,qr =0

0n



z + q1 + q2 + · · · + qr p



(p, r, n ∈ N),

(98)

62

Zeta and q-Zeta Functions and Associated Series and Integrals

where fn (z, p, r) :=

n Y

(−1)m

z (m−1 )−

n

Rn−m+1



Pp−1

q1 ,... ,qr =0

z+q1 +q2 +···+qr p



m−1

o

m=1

and Rm are given as in 2.1(28). Thus, in view of 2.1(16), ζn (0, z) can be expressed in terms of the generalized Bernoulli polynomials of degree n. The special case of (98), when n = 2 and r = 2, reduces to 1.4(21) by letting z = pa. Indeed, starting with p−1 X q1 ,...,qr =0

  q1 + q2 + · · · + qr ζn s, a + p

p−1 X

=

 ∞ X

q1 ,... ,qr =0 k1 , ... , kn ∞ X

= ps

q1 + q2 + · · · + qr a+ + k1 + k2 + · · · + kn p

p−1 X

−s

(pa + q1 + q2 + · · · + qr + p k1 + p k2 + · · · + p kn )−s

k1 , ... , kn q1 ,... ,qr =0 ∞ X

= ps

(pa + k1 + k2 + · · · + kn )−s = ps ζn (s, pa),

k1 , ... , kn

which yields p−1 X q1 ,... ,qr =0

  q1 + q2 + · · · + qr = ps ζn (s, pa). ζn s, a + p

(99)

Differentiating each side of (99) with respect to s and setting s = 0 in the resulting equation, and if we take the exponential in the last identity and then consider 2.1(27) with pa = z, we obtain our desired multiplication formula (98). A special case of (98), when n = 1, yields a multiplication formula for the Gamma function 0 as follows: p−1 Y q1 ,... ,qr =0



z + q1 + q2 + · · · + qr 0 p



= (2π) 2 (p −1) p 2 −z 0(z) 1

r

1

(p, r ∈ N). (100)

Introduction and Preliminaries

63

1.5 The Gaussian Hypergeometric Function and its Generalization The Gauss Hypergeometric Equation The second-order linear ordinary differential equation: z(1 − z)

dw d2 w + [c − (a + b + 1) z] − abw = 0 dz dz2

(1)

or, equivalently, {δ(δ + c − 1) − z(δ + a)(δ + b)}w = 0

  d δ := z , dz

(2)

in which a, b and c are real or complex parameters, is called the Gauss hypergeometric equation. Its only singularities are at z = 0, 1, ∞; each singularity is easily seen to be of regular kind. The hypergeometric equation (1) or (2) is the most celebrated equation of the Fuchsian class, which consists of differential equations, whose only singularities (including the point at infinity) are regular singular points. Its importance stems, in part, from a well-known theorem that every homogeneous linear differential equation of the second order, whose singularities (including the point at infinity) are regular and at most three in number, can be transformed into the hypergeometric equation. Power-series solutions of the hypergeometric equation (1) valid in the neighborhoods of the regular singular points z = 0, 1 or ∞ can be developed by direct application of the classical method of Frobenius. Thus, if c is not an integer, the general solution of (1) valid in a neighborhood of the origin (z = 0) is found to be w = C1 2 F1 (a, b; c; z) + C2 z1−c 2 F1 (a − c + 1, b − c + 1; 2 − c; z)

(c 6∈ Z), (3)

where C1 and C2 are arbitrary constants, and (for convenience) ab a(a + 1)b(b + 1) 2 z+ z + ··· 1·c 1 · 2 · c(c + 1) ∞ X
(a)n (b)n zn = c 6∈ Z− 0 , (c)n n!

2 F1 (a, b; c; z) :=

1+

n=0

in terms of the Pochhammer symbol (λ)n defined by 1.1(5).

(4)

64

Zeta and q-Zeta Functions and Associated Series and Integrals

Gauss’s Hypergeometric Series The infinite series in (4) obviously reduces to the elementary geometric series: ∞ X

zn = 1 + z + z2 + · · · + zn + · · ·

(5)

n=0

in its special cases, when (i)

a=c

and

b=1

or

(ii)

a=1

and

b = c.

(6)

Hence, it is called the hypergeometric series or, more precisely, Gauss’s hypergeometric series after the famous German mathematician, Carl Friedrich Gauss (1777–1855), who in the year 1812 introduced this series into analysis and gave the F–notation for it. (See also Equation 1.5(51) in which the general notation p Fq is introduced, p and q being any nonnegative integers.) By d’Alembert’s ratio test, it is easily seen that the hypergeometric series in (4) converges absolutely within the unit circle, that is, when |z| < 1, provided that the denominator parameter c is neither zero nor a negative integer. Notice, however, that, if either or both of the numerator parameters a and b in (4) is zero or a negative integer, the hypergeometric series terminates, and the question of convergence does not enter into the discussion. Further tests show that the hypergeometric series in (4), when |z| = 1 (that is, on the unit circle), is (i) absolutely convergent, if <(c − a − b) > 0; (ii) conditionally convergent, if −1 < <(c − a − b) 5 0 (iii) divergent, if <(c − a − b) 5 −1.

(z 6= 1);

As a matter of fact, in Case (i), we are led to the well-known Gauss’s summation theorem: 0(c) 0(c − a − b) 0(c − a) 0(c − b)
<(c − a − b) > 0; c 6∈ Z− 0 .

2 F1 (a, b; c; 1) =

(7)

An obvious special case of (7) occurs when the numerator parameter a or b is a nonpositive integer −n. We, thus, have the Chu-Vandermonde summation formula: 2 F1 (−n, b; c; 1) =

(c − b)n (c)n


n ∈ N0 ; c 6∈ Z− 0 ,

(8)

which is, in fact, equivalent to Vandermonde’s convolution theorem:    X   n   n  X λ µ λ+µ λ µ = = k n−k n n−k k k=0

k=0

(n ∈ N0 ; λ, µ ∈ C).

(9)

Introduction and Preliminaries

65

For a number of summation theorems for the hypergeometric series (4), when z takes on other special values, see Bailey [87, pp. 9–11], Erde´ lyi et al. [421, pp. 104– 105], Slater [1040, p. 243] and Luke [779, pp. 271–273].

The Hypergeometric Series and Its Analytic Continuation We have seen that the hypergeometric series in (4) converges absolutely, when |z| < 1 and, thus, defines a function: 2 F1 (a, b; c; z),

which is analytic, when |z| < 1, provided that c is neither zero nor a negative integer. This function is correspondingly called the hypergeometric function or Gauss’s hypergeometric function. Indeed, it is the only solution of the Fuchsian equation (1) that is analytic at the point z = 0 and assumes the value 1 at that point. The hypergeometric function 2 F1 (a, b; c; z) can, in fact, be continued analytically outside the unit circle in a number of ways. If, for convenience, we use this same notation for the analytically continued function, one way is to employ Euler’s integral representation: 0(c) 2 F1 (a, b; c; z) = 0(a) 0(c − a)

Z1

ta−1 (1 − t)c−a−1 (1 − zt)−b dt

(10)

0

(<(c) > <(a) > 0; | arg(1 − z)| 5 π −  (0 <  < π)) or, equivalently, 0(c) 2 F1 (a, b; c; z) = 0(b) 0(c − b)

Z1

tb−1 (1 − t)c−b−1 (1 − zt)−a dt

(11)

0

(<(c) > <(b) > 0; | arg(1 − z)| 5 π −  (0 <  < π)), since, by the series definition (4), 2 F1 (a, b; c; z) ≡ 2 F1 (b, a; c; z).

(12)

An analytic continuation without such parametric constraints as in (10) and (11) can be provided by employing the following Mellin-Barnes contour integral representation of the hypergeometric function: 0(a) 0(b) 2 F1 (a, b; c; z) 0(c) Zi∞ 1 0(a + ζ ) 0(b + ζ ) 0(−ζ ) (−z)ζ dζ = 2πi 0(c + ζ ) −i∞


| arg(−z)| 5 π −  (0 <  < π); a, b 6∈ Z− 0 ,

(13)

66

Zeta and q-Zeta Functions and Associated Series and Integrals

where the path of integration is the imaginary axis (in the complex ζ -plane), which is indented, if necessary, to ensure that the poles of 0(a + ζ ) 0(b + ζ ), viz ζ = −a − n

and ζ = −b − n (n ∈ N0 ),

(14)

lie to the left of the path and the poles of 0(−ζ ), viz ζ = 0, 1, 2, . . . ,

(15)

lie to the right of the path. The integrals in (10), (11) and (13) define analytic functions of z, which are singlevalued in the domain | arg(−z)| < π, that is, in the whole z-plane, with the exception of the points on the negative real axis. Since the Gaussian function 2 F1 (a, b; c; z) is defined by the hypergeometric series (4) everywhere within the unit circle |z| = 1 (including, for example, at points on the negative real axis from z = 0 to z = 1 − ,  being an arbitrarily small positive number), we can use either of these integrals to provide the analytic continuation of the hypergeometric function to the whole complex z-plane cut along the real axis from z = 1 to z = ∞. This analytic continuation of 2 F1 (a, b; c; z) is conveniently denoted by the same symbol 2 F1 (a, b; c; z), which will henceforth represent a branch (the principal branch) of the analytic function generated by the Gaussian hypergeometric series in (4). By evaluating the contour integral in (13) using the sum of the residues of the integrand at the poles given by (14), it can be shown that 1 2 F1 (a, b; c; z) 0(c)   0(b − a) 1 = (−z)−a 2 F1 a, 1 − c + a; 1 − b + a; 0(b) 0(c − a) z (16)   1 0(a − b) (−z)−b 2 F1 b, 1 − c + b; 1 − a + b; + 0(a) 0(c − b) z (| arg(−z)| 5 π −  (0 <  < π); a − b 6∈ Z). This is an important result, for it may be used to provide the analytic continuation of 2 F1 (a, b; c; z) into the domain |z| > 1, that is, outside the circle of convergence of the hypergeometric series (4). It also shows that the function so defined possesses a branch point at infinity and that, asymptotically, −a + B (−z)−b 2 F1 (a, b; c; z) ∼ A (−z)

(|z| → ∞; | arg(−z)| 5 π −  (0 <  < π)),

(17)

where, for convenience, A :=

0(c) 0(b − a) 0(b) 0(c − a)

and B :=

0(c) 0(a − b) . 0(a) 0(c − b)

(18)

Introduction and Preliminaries

67

Linear, Quadratic and Cubic Transformations Pfaff-Kummer transformations:  z z−1
− c 6∈ Z0 ; |arg(1 − z)| 5 π −  (0 <  < π) ;   z −b F b, c − a; c; F (a, b; c; z) = (1 − z) 2 1 2 1 z−1  −a F (a, b; c; z) = (1 − z) F 2 1 2 1 a, c − b; c;

(19)

(20)

(c 6∈ Z− 0 ; |arg(1 − z)| 5 π −  (0 <  < π)). Euler’s transformation: c−a−b 2 F1 (a, b; c; z) = (1 − z) 2 F1 (c − a, c − b; c; z) − (c 6∈ Z0 ; |arg(1 − z)| 5 π −  (0 <  < π )).

(21)

The (Pfaff-Kummer) transformations (19) and (20) can be proven, for example, by setting t = 1 − s in Euler’s integral representations (10) and (11), respectively, and then appealing to the principle of analytic continuation. The Euler transformation (21) follows, if we perform the transformations (19) and (20) successively. For an extensive list of quadratic and cubic transformations of the Gaussian hypergeometric function, see Erde´ lyi et al. [421, pp. 110–114] and Luke [779, Vol. I, p. 92 et seq.].

Hypergeometric Representations of Elementary Functions

(1 − z)−a =

∞ X n=0

(a)n

zn = 2 F1 (a, b; b; z); n!

log(1 + z) = z 2 F1 (1, 1; 2; −z);     1+z 3 1 log = 2z 2 F1 , 1; ; z2 ; 1−z 2 2   1 1 3 2 , ; ;z ; sin −1 z = z 2 F1 2 2 2   1 3 −1 2 tan z = z 2 F1 , 1; ; −z ; 2 2   p sinh −1 z = log z + 1 + z2   1 1 3 2 = z 2 F1 , ; ; −z ; 2 2 2

(22) (23) (24) (25) (26)

(27)

68

Zeta and q-Zeta Functions and Associated Series and Integrals

    1−2a √ 1 1 = 2 F1 a, a − ; 2a; z 1+ 1−z 2 2   √ 1 = 1 − z 2 F1 a, a + ; 2a; z ; 2   √
−2a √
−2a 1 1 + 1− z = 2 2 F1 a, a + ; ; z . 1+ z 2 2

(28)

(29)

Hypergeometric Representations of Other Functions Legendre functions of the first and second kind:   1  1−z 1 z + 1 2µ ; 2 F1 −ν, ν + 1; 1 − µ; 0(1 − µ) z − 1 2
1 √ iµπ 2 − 1 2µ z π e 0(µ + ν + 1)   Qµ ν (z) = 2ν+1 zµ+ν+1 0 ν + 32   1 3 1 (µ + ν + 1), (µ + ν) + 1; ν + ; z−2 . · 2 F1 2 2 2 Pµ ν (z) =

(30)

(31)

The incomplete Beta function: Bz (α, β) = α −1 zα 2 F1 (α, 1 − β; α + 1; z) .

(32)

Complete Elliptic integrals of the first and second kind: Zπ/2 K(k) = 0

dθ 1 = π 2 F1 √ 2 1 − k2 sin 2 θ



 1 1 , ; 1; k2 ; 2 2

  Zπ/2p 1 1 1 2 2 2 1 − k sin θ dθ = π 2 F1 − , ; 1; k . E(k) = 2 2 2

(33)

(34)

0

Jacobi polynomials: P(α,β) (z) = (−1)n P(β,α) (−z) n n # "   −n, α + β + n + 1; 1 − z α+n . = 2 F1 n α + 1; 2

(35)

Introduction and Preliminaries

69

Gegenbauer (or ultraspherical) polynomials:  −1    1 1 ν− 2 ,ν− 2 n + ν − 12 n + 2ν − 1 (z) Pn n n     −n, 2ν + n; n + 2ν − 1 1 − z  F = . 2 1 1 n ν+ ; 2 2

Cnν (z) =

(36)

Legendre (or spherical) polynomials:   1 1−z 2 Pn (z) = P(0,0) (z) = C (z) = F −n, n + 1; 1; . 2 1 n n 2

(37)

Tchebycheff polynomials of the first and second kind: n − 12 Tn (z) = n 

−1



Pn

− 12 ,− 12



(z) =

  1 1 1−z n Cn0 (z) = 2 F1 −n, n; ; , 2 2 2

(38)

where n o Cn0 (z) := lim ν −1 Cnν (z) ;

(39)

ν→0

 −1 1 1 1 n + 12 2,2 Pn (z) = Cn1 (z) 2 n+1   3 1−z . = (n + 1) 2 F1 −n, n + 2; ; 2 2 



Un (z) =

(40)

The Confluent Hypergeometric Function If, in Gauss’s hypergeometric equation (1), we replace z by z/b, the resulting equation will have three regular singularities at z = 0, b, ∞. By letting |b| → ∞, this transformed equation reduces to the form: z

d2 w dw + (c − z) − aw = 0 2 dz dz

(41)

or, equivalently,  {δ(δ + c − 1) − z(δ + a)} w = 0

 d δ := z . dz

(42)

70

Zeta and q-Zeta Functions and Associated Series and Integrals

Equation (41) or (42) has a regular singularity at z = 0 and an irregular singularity at z = ∞, which is formed by the confluence of two regular singularities at b and ∞ of Gauss’s equation (1) with z replaced by z/b. Consequently, (41) or (42) is called the confluent hypergeometric equation or Kummer’s differential equation after Ernst Eduard Kummer (1810–1893), who presented a detailed study of its solutions in 1836 (cf. Kummer [707]). The simplest solution of (1) or (2) is Kummer’s function F(a, c, z), which, in the notations of this section, is 1 F1 (a; c; z) := 1 +

a(a + 1) 2 a z+ z + ··· 1·c 1 · 2 · c(c + 1)

∞ X (a)n zn = (c)n n!

(c 6∈ Z− 0;

(43)

|z| < ∞).

n=0

Moreover, since   n  z n o (µ z)n (λ)n = zn = lim |λ|→∞ |µ|→∞ λ (µ)n (n ∈ N0 ; |z| < ∞), lim

(44)

we readily have 1 F1 (a; c; z) =

lim

|b|→∞

2 F1

 z a, b; c; . b

(45)

In view of the principle of confluence involved in (45), the solution (43) is also called the confluent hypergeometric function. Other popular notations for the series solution (43) are (i) M(a, c, z), used by Airey and Webb [15], and (ii) 8(a; c; z) or 8(a, c, z), introduced by Humbert [573, 574].

Important Properties of Kummer’s Confluent Hypergeometric Function The principle of confluence exhibited by (45) is useful to deduce properties of the confluent hypergeometric 1 F1 function from those of Gauss’s hypergeometric 2 F1 function. Thus, from (10) and (20), we obtain 0(c) 1 F1 (a; c; z) = 0(a) 0(c − a)

Z1

ta−1 (1 − t)c−a−1 ezt dt

(46)

0

(<(c) > <(a) > 0); z 1 F1 (a; c; z) = e 1 F1 (c − a; c; −z),

(47)

Introduction and Preliminaries

71

which is known as Kummer’s first transformation. We also have Kummer’s second transformation: −z

e

  1 1 2 1 F1 (a; 2a; 2z) = 0 F1 −; a + ; z 2 4 (2a 6= −1, −3, −5, . . .),

(48)

where 0 F1

1 F1 (a; c; z/a) |a|→∞ ∞ X zn = (c 6∈ Z− 0; n! (c)n n=0

(−; c; z) = lim

|z| < ∞).

(49)

It follows immediately from the definition (43) that z 1 F1 (a; a; z) = 0 F0 (−; −; z) = e .

(50)

The Generalized (Gauss and Kummer) Hypergeometric Function A natural generalization of the hypergeometric functions 2 F1 , 1 F1 , et cetera (considered in this section) is accomplished by the introduction of an arbitrary number of numerator and denominator parameters. The resulting series: " p Fq

# ∞ X (α1 )n · · · (αp )n zn α1 , . . . , αp ; z = (β1 )n · · · (βq )n n! β1 , . . . , βq ; n=0

(51)

= p Fq (α1 , . . . , αp ; β1 , . . . , βq ; z), where (λ)n is the Pochhammer symbol defined by 1.1(5), is known as the generalized Gauss (and Kummer) series, or simply, the generalized hypergeometric series. Here, p and q are positive integers or zero (interpreting an empty product as 1), and we assume (for simplicity) that the variable z, the numerator parameters α1 , · · ·, αp and the denominator parameters β1 , . . ., βq take on complex values, provided that no zeros appear in the denominator of (51), that is, that (βj 6∈ Z− 0 ; j = 1, . . ., q).

(52)

Thus, if a numerator parameter is a negative integer or zero, the p Fq series terminates in view of the identity 1.1(47), and we are led to a (generalized hypergeometric)

72

Zeta and q-Zeta Functions and Associated Series and Integrals

polynomial of the type: " p+1 Fq

# n X −n, α1 , . . . , αp ; (−n)k (α1 )k . . . (αp )k zk z = (β1 )k . . . (βq )k k! β1 , . . . , βq ; k=0

=

(α1 )n . . . (αp )n (−z)n (β1 )n . . . (βq )n " · q+1 Fp

−n, 1 − β1 − n, . . . , 1 − βq − n; (−1)p+q z 1 − α1 − n, . . . , 1 − αp − n;

(53) # (n ∈ N0 ),

where we have reversed the order of the terms of the polynomial by using 1.1(25) and 1.1(26). Assuming that none of the numerator parameters is zero or a negative integer (otherwise the question of convergence will not arise) and with the usual restriction given by (52), the p Fq series in (51) (i) converges for |z| < ∞, if p ≤ q, (ii) converges for |z| < 1, if p = q + 1, and (iii) diverges for all z, z 6= 0, if p > q + 1.

Furthermore, if we set ω :=

q X j=1

βj −

p X

αj ,

(54)

j=1

then it is known that the p Fq series in (51), with p = q + 1, is (i) absolutely convergent for |z| = 1, if <(ω) > 0, (ii) conditionally convergent for |z| = 1 (z 6= 1), if −1 < <(ω) 5 0, and (iii) divergent for |z| = 1, if <(ω) 5 −1.

Analytic Continuation of the Generalized Hypergeometric Function The generalized hypergeometric p Fq function is defined by the (absolutely) convergent series (51), whenever (i) p 5 q and |z| < ∞

or (ii) p = q + 1 and |z| < 1.

As in the above-detailed case of the 2 F1 function, the generalized hypergeometric function p Fq (with p = q + 1) can be continued analytically to the domain | arg(1 − z) | < π, that is, to the whole complex z-plane cut along the real axis from z = 1 to

Introduction and Preliminaries

73

z = ∞, by using (51) in conjunction with the Mellin–Barnes contour integral representation: " # α1 , . . . , αp ; 0(α1 ) . . . 0(αp ) z p Fq 0(β1 ) . . . 0(βq ) β1 , . . . , βq ;

=

Zi∞

1 2πi

−i∞

0(α1 + ζ ) . . . 0(αp + ζ ) 0(−ζ ) (−z)ζ dζ 0(β1 + ζ ) . . . 0(βq + ζ )

(55)

(αj 6∈ Z− 0 ; j = 1, . . . , p; | arg(−z)| < π), where the path of integration is the imaginary axis (in the complex ζ -plane), which is indented, if necessary, to separate the poles of 0(αj + ζ ) (j = 1, · · · , p) from those of 0(−ζ ).

Functions Expressible in Terms of the p Fq Function The following list is in addition to the important functions (considered above in this section) that can be expressed as a hypergeometric 2 F1 function. Elementary functions: ez = 0 F0 [−; −; z];

(56)

(1 − z)

(57)

−α

= 1 F0 [α; −; z];   1 2 1 cos z = 0 F1 −; ; − z ; 2 4   3 1 2 sin z = z 0 F1 −; ; − z . 2 4

(58) (59)

Bessel functions: 

1 2z

ν

 1 −; ν + 1; − z2 0(ν + 1) 4  ν 1   2z 1 = exp(±iz) 1 F1 ν + ; 2ν + 1; ∓2iz ; 0(ν + 1) 2

Jν (z) =



1 2z



0 F1

(60)

ν 

 1 2 Iν (z) = 0 F1 −; ν + 1; z 0(ν + 1) 4  ν 1   2z 1 = exp(±z) 1 F1 ν + ; 2ν + 1; ∓2z . 0(ν + 1) 2

(61)

74

Zeta and q-Zeta Functions and Associated Series and Integrals

Lommel functions: sµ,ν (z) =

zµ+1 (µ − ν + 1)(µ + ν + 1) 

 1 · 1 F2  1 − z2  ; 1 (µ − ν + 3), (µ + ν + 3); 4 2 2     µ+ν +1 µ − ν +1 µ−1 0 Sµ,ν (z) = sµ,ν (z) + 2 0 2 2       1 1 · sin (µ − ν)π Jν (z) − cos (µ − ν)π Yν (z) , 2 2 1;

(62)

(63)

where Jν (z) is given by (60), and Yν (z) = csc(νπ) [cos (νπ)Jν (z) − J−ν (z)] .

(64)

Struve functions: 

ν+1

 1; 1 2  1 F2  3 Hν (z) =    3 − z  3 3 ,ν+ ; 4 0 2 0 ν+2 2 2 1−ν 2  sν,ν (z) =√  π 0 ν + 12 1 2z



21−ν  Sν,ν (z); = Yν (z) + √  π 0 ν + 12  ν+1   1 1; 2z 1 2  1 F2  3 Lν (z) =    3 z  3 3 4 , ν + ; 0 2 0 ν+2 2 2   1 = exp − (ν + 1)iπ Hν (iz). 2 Whittaker function of the first kind:     1 1 µ+ 12 Mκ,µ (z) = z exp − z 1 F1 µ − κ + ; 2µ + 1; z 2 2     1 1 1 µ+ 2 =z exp z 1 F1 µ + κ + ; 2µ + 1; −z . 2 2

(65)

(66)

(67)

Incomplete Gamma function: γ (α, z) = α −1 zα 1 F1 (α; α + 1; −z) .

(68)

Introduction and Preliminaries

75

Error function: √   1 1 2 π erf(z) = γ ,z 2 2 2   1 3 ; ; −z2 . = z 1 F1 2 2

Erf(z) =

(69)

Hermite polynomials: [n/2] X

(−1)k n! (2z)n−2k k! (n − 2k)! k=0   1 1 1 = (2z)n 2 F0 − n, − n; −; −z−2 2 2 2

Hn (z) =

(n ∈ N0 ).

(70)

Laguerre functions and polynomials: Lν(α) (z) = Ln(α) (z) =

0(ν + α + 1) 1 F1 (−ν; α + 1; z) ; 0(ν + 1) 0(α + 1)  n  X n + α (−z)k

n−k k! k=0   n+α = 1 F1 (−n; α + 1; z) n   1 (−z)n = 2 F0 −n, −α − n; −; − n! z

(n ∈ N0 ).

(71)

(72)

Poisson–Charlier polynomials: cn (x; α) =

n X k=0

   n x (−1) k! α −k k k k

(α > 0; x ∈ N0 )

= n! (−α)−n Ln(x−n) (α)   x = (−α)−n n! 1 F1 (−n; x − n + 1; α) . n

(73)

In an alternative notation, we have (see Szego¨ [1141, p. 35]) (x − n + 1)n √ 1 F1 (−n; x − n + 1; α) n!   1 √ x = α − 2 n n! 1 F1 (−n; x − n + 1; α) . n 1

pn (x) = α − 2 n

(74)

76

Zeta and q-Zeta Functions and Associated Series and Integrals

1.6 Stirling Numbers of the First and Second Kind Stirling Numbers of the First Kind The Stirling numbers s(n, k) of the first kind are defined by the generating functions: z(z − 1) · · · (z − n + 1) =

n X

s(n, k) zk

(1)

zn n!

(2)

k=0

and {log(1 + z)}k = k!

∞ X

s(n, k)

n=k

(|z| < 1).

We have the following recurrence relations satisfied by s(n, k): s(n + 1, k) = s(n, k − 1) − n s(n, k)

(n = k = 1);

  n−j   X k n s(n, k) = s(n − l, j) s(l, k − j) j l

(n = k = j).

(3)

(4)

l=k−j

From the definition (1) of s(n, k), the Pochhammer symbol in 1.1(5) can be written in the form: (z)n = z(z + 1) · · · (z + n − 1) =

n X

(−1)n+k s(n, k) zk ,

(5)

k=0

where (−1)n+k s(n, k) denotes the number of permutations of n symbols, which has exactly k cycles. It is not difficult to see also that ( 1 (n = 0) s(n, n) = 1, s(n, 0) = 0 (n ∈ N), (6)   n n+1 s(n, 1) = (−1) (n − 1)!, s(n, n − 1) = − 2 and n X

s(n, k) = 0 (n ∈ N\{1});

k=1 n X j=k

n X k=0

s(n + 1, j + 1) n

j−k

= s(n, k).

(−1)n+k s(n, k) = n!; (7)

Introduction and Preliminaries

77

Yet another recursion formula for s(n, k) is given by (see Shen [1024]): (k − 1) s(n, k) = −

k−1 X

(m)

s(n, k − m) Hn−1 ,

(8)

m=1 (s)

where Hn is the generalized harmonic numbers of order s, defined by Hn(s) :=

n X 1 ks

(n ∈ N; s ∈ C)

(9)

k=1

(1)

and Hn := Hn (n ∈ N) is the harmonic numbers. (0) (m) Here, for (8), we assume H0 := 1 and H0 := 0 (m ∈ N). It readily follows from the recursion formula (8) that s(n, 2) = (−1)n (n − 1)! Hn−1 ; i (n − 1)! h (2) s(n, 3) = (−1)n+1 (Hn−1 )2 − Hn−1 ; 2 h i (n − 1)! (2) (3) s(n, 4) = (−1)n (10) (Hn−1 )3 − 3Hn−1 Hn−1 + 2Hn−1 ; 6 (n − 1)! s(n, 5) = (−1)n+1 24     (3) (2) 2 (4) 4 2 (2) · (Hn−1 ) + 8 Hn−1 Hn−1 − 6 (Hn−1 ) Hn−1 + 3 Hn−1 − 6 Hn−1 . In view of (5), by logarithmically differentiating 1.1(19) and then using 1.2(7), we obtain n

X d {(z)n } = (−1)n+k k s(n, k) zk−1 dz k=1 ! n X 1 = (z)n , z+k−1

(11)

k=1

which, upon employing Leibniz’s rule for differentiation, yields a more general formula: n X dj+1 {(z) } = (−1)n+k+j+1 (−k)j+1 s(n, k) zk−j−1 n dzj+1 k=j+1 !  j n X X j! 1 dj−l l {(z)n } = (−1) (j − l)! (z + k − 1)l+1 dzj−l l=0

(j ∈ N0 ; n ∈ N).

k=1

(12)

78

Zeta and q-Zeta Functions and Associated Series and Integrals

For j = 1 and j = 2, (12) immediately yields n X

(−1)n+k k(k − 1) s(n, k) zk−2

k=2

 = (z)n 

n X k=1

1 z+k−1

!2 −

n X k=1

(13)

 1  (z + k − 1)2

and n X

(−1)n+k k(k − 1)(k − 2) s(n, k) zk−3

k=3

 = (z)n 

n X k=1

+

1 z+k−1

n X k=1

!3 −3

n X k=1

1 z+k−1

!

n X k=1

1 (z + k − 1)2

! (14)

# 1 . (z + k − 1)3

Stirling Numbers of the Second Kind The Stirling numbers S(n, k) of the second kind are defined by the generating functions: n

z =

n X

S(n, k) z(z − 1) · · · (z − k + 1),

(15)

k=0

(ez − 1)k = k!

∞ X n=k

S(n, k)

zn , n!

(16)

and (1 − z)−1 (1 − 2z)−1 · · · (1 − kz)−1 =

∞ X

S(n, k) zn−k

(|z| < k−1 ),

(17)

n=k

where S(n, k) denotes the number of ways of partitioning a set of n elements into k nonempty subsets. It is not difficult to see also that ( 1 (n = 0) S(n, 0) = 0 (n ∈ N),   n S(n, 1) = S(n, n) = 1, and S(n, n − 1) = . (18) 2

Introduction and Preliminaries

79

The recurrence relations for S(n, k) are given by (n = k = 1)

S(n + 1, k) = k S(n, k) + S(n, k − 1)

(19)

and   n−j   X k n S(n, k) = S(n − i, j) S(i, k − j) j i

(n = k = j).

(20)

i=k−j

The numbers S(n, k) can be expressed in an explicit form:

S(n, k) =

  k k n 1 X j . (−1)k−j k! j

(21)

j=0

Some additional properties of S(n, k) are recalled here as follows: n X k=0 n X

(−1)n−k k! S(n, k) = 1;

(22)

S(j − 1, k − 1) kn−j = S(n, k);

(23)

j=k

S(n, k) =

n−k X

(−1)

j



j=0 n X

n−1+j n−k+j



 2n − k S(n − k + j, j); n−k−j

S(j, k) S(n, j) = δkn ,

(24)

(25)

j=k

where δmn denotes the Kronecker delta defined by ( δmn =

0 (m 6= n), 1 (m = n).

( and δm =

0 (m 6= 0), 1 (m = 0).

(26)

Relationships Among Stirling Numbers of the First and Second Kind and Bernoulli Numbers Akiyama and Tanigawa [17] presented, to evaluate multiple zeta values at nonpositive integers, the following identities:   n X 1 n 1 S(n, `) s(`, k) = Bn−k + δn−k−1 ` n k `=k

(n, k ∈ N; n = k),

(27)

80

Zeta and q-Zeta Functions and Associated Series and Integrals

where Bk is the Bernoulli number given in 1.6. n   X n

n S(n − 1, k − 1) (n, k ∈ N). k `=0   n X S(n, `) s(` + 1, k) Bn+1−k n + 1 = (n, k ∈ N; n = k − 1), `+1 n+1 k `

Bn−` S(`, k) =

(28)

(29)

`=k−1

which, upon setting k = 1 and using (21), yields

Bn =

n X (−1)` `! S(n, `) `+1 `=0

=

n X `=0

  ` ` n 1 X (−1) j j `+1 j

(n ∈ N0 ) ,

(30)

j=0

or, equivalently, n X

s(n, `) B` =

`=0

(−1)n n! n+1

(n ∈ N0 ) .

(31)

We also recall some known formulas (see, e.g., [982]): max{k, Xj}+1

s(`, j) S(k, `) = δjk .

(32)

s(k, `) S(`, j) = δjk .

(33)

`=0 max{k, Xj}+1 `=0

s(n, i) =

n X k X

s(n, k) s(k, j) S(j, i).

(34)

S(n, k) S(k, j) s(j, i).

(35)

k=i j=0

S(n, i) =

n X k X k=i j=0 n−m X

  k+n−1 2n − m S(n, m) = (−1) s(k − m + n, k). k+n−m n−k−m k=0    n−m X 2n − m k k+n−1 s(n, m) = (−1) S(k − m + n, k). k+n−m n−k−m k=0

k



(36)

(37)

Introduction and Preliminaries

81

1.7 Bernoulli, Euler and Genocchi Polynomials and Numbers Bernoulli Polynomials and Numbers The Bernoulli polynomials Bn (x) are defined by the generating function: ∞

X z exz zn = B (x) n ez − 1 n!

(|z| < 2π).

(1)

n=0

The numbers Bn := Bn (0) are called the Bernoulli numbers generated by ∞

X z zn = Bn z e −1 n!

(|z| < 2π).

(2)

n=0

It easily follows from (1) and (2) that Bn (x) =

n   X n

k

k=0

Bk xn−k .

(3)

The Bernoulli polynomials Bn (x) satisfy the difference equation: Bn (x + 1) − Bn (x) = n xn−1

(n ∈ N0 ),

(4)

which yields Bn (0) = Bn (1)

(n ∈ N \ {1}).

(5)

Setting x = 1 in (3), in view of (5), we have Bn =

n   X n k=0

k

Bk ,

(6)

which gives a recursion formula for computing Bernoulli numbers. The first few of the Bernoulli numbers are already listed with the Euler-Maclaurin summation formula 1.4(68), and (for the sake of completeness) we have the following list:

B0 = 1,

1 B2 = , 6

691 , 2730

7 B14 = , 6

854513 , 138

B24 = −

B12 = − B22 =

1 B1 = − , 2

B4 = −

B16 = −

1 , 30

3617 , 510

236364091 , 2730

B6 = B18 =

B26 =

1 , 42

1 , 30

B10 =

B20 = −

174611 , 330

B8 = −

43867 , 798

8553103 , . . . , B2n+1 = 0 6

5 , 66

(n ∈ N).

(7)

82

Zeta and q-Zeta Functions and Associated Series and Integrals

The first few of the Bernoulli polynomials are given below: 1 1 B1 (x) = x − , B2 (x) = x2 − x + , 2 6 3 2 1 1 3 4 3 2 B3 (x) = x − x + x, B4 (x) = x − 2x + x − , 2 2 30 5 5 1 B5 (x) = x5 − x4 + x3 − x, 2 3 6 1 1 5 B6 (x) = x6 − 3x5 + x4 − x2 + , 2 2 42 7 6 7 5 7 3 1 7 B7 (x) = x − x + x − x + x, . . . . 2 2 6 6 B0 (x) = 1,

(8)

It is not difficult to derive the following identities for the Bernoulli polynomials: B0n (x) = n Bn−1 (x) (n ∈ N); Bn (1 − x) = (−1)n Bn (x) (n ∈ N0 );

(9) (10)

(−1)n Bn (−x) = Bn (x) + n xn−1

(11)

(n ∈ N0 ).

Multiplication formula:

Bn (mx) = m

n−1

m−1 X

 Bn

k=0

k x+ m



(n ∈ N0 , m ∈ N).

(12)

Addition formula: Bn (x + y) =

n   X n k=0

k

Bk (x) yn−k

(n ∈ N0 ).

(13)

Integral formulas: Zy x

Bn (t) dt =

Bn+1 (y) − Bn+1 (x) ; n+1

(14)

Zx+1 Bn (t) dt = xn ;

(15)

x

Z1 0

Bn (t) Bm (t) dt = (−1)n−1

m! n! Bm+n (m + n)!

(m, n ∈ N).

(16)

Introduction and Preliminaries

83

It follows from (14) and (15) that the finite sum of powers is expressed as Bernoulli polynomials and numbers: m X

kn =

k=1

Bn+1 (m + 1) − Bn+1 n+1

(m, n ∈ N).

(17)

By writing 2n for n in (3), we can deduce that B2n (x) + n x

2n−1

=

n   X 2n

2k

k=0

B2k x2n−2k ,

which, upon integrating from 0 to 12 , yields n X k=0

1 22k B2k = (2k)!(2n − 2k + 1)! (2n)!

(n ∈ N0 ),

(18)

where we have applied (14). It is readily shown that B2n

    1 = 21−2n − 1 B2n 2

and B2n+1

  1 = 0 (n ∈ N). 2

(19)

The Generalized Bernoulli Polynomials and Numbers (α)

The generalized Bernoulli polynomials Bn (x) of degree n in x are defined by the generating function: 

z z e −1

α

exz =

∞ X

B(α) n (x)

n=0

zn n!


|z| < 2π; 1α := 1

(20)

for arbitrary (real or complex) parameter α. Clearly, we have n (α) B(α) n (x) = (−1) Bn (α − x),

(21)

so that n (α) n (α) B(α) n (α) = (−1) Bn (0) =: (−1) Bn ,

(22) (α)

in terms of the generalized Bernoulli numbers Bn defined by the generating function: 

z z e −1

α =

∞ X n=0

B(α) n

zn n!

(|z| < 2π ; 1α := 1).

(23)

84

Zeta and q-Zeta Functions and Associated Series and Integrals

It is easily observed that (1) B(1) n (x) = Bn (x) and Bn = Bn

(n ∈ N0 ).

(24)

From the generating function (20), it is fairly straightforward to deduce the addition theorem: B(α+β) (x + y) = n

n   X n

k

k=0

(β)

(α)

Bk (x) Bn−k (y),

(25)

which, for x = β = 0, corresponds to the elegant representation: B(α) n (x) =

n   X n k=0

k

(α)

Bk xn−k

(26)

for the generalized Bernoulli polynomials as a finite sum of the generalized Bernoulli numbers. Srivastava et al. [1101, p. 442, Eqs. (4.4) and (4.5)] gave two new classes of addition theorems for the generalized Bernoulli polynomials: ) B(α+λγ (x + γ y) = n

  n X γ + n n (α−λk) (λk+λγ ) B (x − ky) Bn−k (ky + γ y) γ +k k k

(<(γ ) > 0);

k=0

(27) Bn(α+β+n+1) (x + y + n) =

n  X k=0



n (α+k+1) (β+n−k+1) B (x + k) Bn−k (y + n − k). (28) k k

Srivastava and Todorov [1110, p. 510, Eq. (3)] proved the following explicit formula for the generalized Bernoulli polynomials: B(α) n (x) =

   k n   X n α+k−1 k! X j k 2k (−1) j (x + j)n−k (2k)! j k k j=0

k=0

· 2 F1 [k − n, k − α; 2k + 1; j/(x + j)],

(29)

in terms of the Gaussian hypergeometric function (see Section 1.5). They also applied the representation (29) to derive certain interesting special cases considered earlier by Gould [499] and Todorov [1153]. Indeed, by the Chu-Vandermonde theorem 1.5(9), we have 2 F1 (−N, b; c; 1) =



  c − b + N − 1 c + N − 1 −1 N N

(N ∈ N0 ),

Introduction and Preliminaries

85

which, for N = n − k, b = k − α and c = 2k + 1, readily yields 2 F1 (k − n, k − α; 2k + 1; 1) =



 α + n (n − k)! (2k)! (n + k)! n−k

(0 ≤ k ≤ n).

(30)

In view of (30), the special case of the Srivastava-Todorov formula (29), when x = 0, gives the following representation for the generalized Bernoulli numbers: B(α) n

=

  n  X α+n α+k−1 n−k

k=0

k

  k X n! j k n+k j (−1) (n + k)! j

(31)

j=0

or, equivalently, B(α) n =

n X

(−1)k

k=0

   α+n α+k−1 n! ∆k 0n+k , n−k k (n + k)!

(32)

where, for convenience, ∆ a =∆ x k r

k r

= x=a

k X

(−1)

j=0

k−j

  k (a + j)r , j

(33)

∆ being the difference operator defined by (cf. Comtet [337, p. 13 et seq.]) ∆ f (x) = f (x + 1) − f (x),

(34)

so that, in general, ∆ f (x) = k

k X

(−1)

k−j

j=0

  k f (x + j). j

(35)

Alternatively, since (Comtet [337, p. 204, Theorem A]; see also Eq. 1.5(20)) S(n, k) =

1 k n ∆ 0 , k!

(36)

where S(n, k) denotes the Stirling number of the second kind defined by 1.6(14), that is, by n

z =

n   X z k=0

k

k! S(n, k),

(37)

86

Zeta and q-Zeta Functions and Associated Series and Integrals

the representation (31) or (32) can be written also as (Todorov [1153, p. 665, Eq. (3)]) B(α) n

=

n X k=0

    α + n α + k − 1 n + k −1 (−1) S(n + k, k). n−k k k k

(38)

Formula (38) provides an interesting generalization of the following known result for the Bernoulli numbers Bn : Bn =

n X

(−1)k



k=0

  n + 1 n + k −1 ∆k 0n+k , k! k+1 k

(39)

which was considered, for example, by Gould [499, p. 49, Eq. (17)].

Euler Polynomials and Numbers The Euler polynomials En (x) and the Euler numbers En are defined by the following generating functions: ∞

X 2exz zn = E (x) n ez + 1 n!

(|z| < π)

(40)

n=0

and ∞

X 2ez zn = sech z = E n n! e2z + 1 n=0

 π |z| < , 2

(41)

respectively. The following formulas are readily derivable from (40) and (41): En (x + 1) + En (x) = 2 xn (n ∈ N0 ); En0 (x) = n En−1 (x) (n ∈ N); En (1 − x) = (−1)n En (x) (n ∈ N0 );

(42) (43) (44)

(−1)n+1 En (−x) = En (x) − 2 xn (n ∈ N0 ); n   X n En (x + y) = Ek (x) yn−k (n ∈ N0 ); k k=0   n   X n Ek 1 n−k x − En (x) = (n ∈ N0 ), k 2k 2

(45) (46)

(47)

k=0

which, upon taking x = 21 , yields En = 2n En

  1 2

(n ∈ N0 );

(48)

Introduction and Preliminaries n   X 2n k=0

2k

87

E2k = 0 (n ∈ N).

(49)

Multiplication formulas: En (mx) = m

n

m−1 X

(−1) En k

k=0



k x+ m



(n ∈ N0 ; m = 1, 3, 5, . . .);

  m−1 X 2 k n k En (mx) = − (−1) Bn+1 x + m n+1 m

(50)

(n ∈ N0 ; m = 2, 4, 6, . . .).

k=0

(51) Integral formulas: Zy

En (t) dt =

x

Z1

En+1 (y) − En+1 (x) n+1

(m, n ∈ N0 );

  En (t) Em (t) dt = (−1)n 4 2m+n+2 − 1

0

(52)

m! n! Bm+n+2 (m + n + 2)!

(m, n ∈ N0 ). (53)

An alternating finite sum of powers can be expressed as the Euler polynomials: m X

(−1)m−k kn =

k=1

 1 En (m + 1) + (−1)m En (0) 2

(m, n ∈ N).

(54)

Fourier Series Expansions of Bernoulli and Euler Polynomials By employing suitable contour integrations in complex function theory, we can obtain the following Fourier series expansions of Bernoulli and Euler polynomials: ∞

B2n (x) =

(−1)n−1 2 (2n)! X cos 2kπx (2π)2n k2n

(n ∈ N; 0 5 x 5 1); (55)

k=1

∞

B2n−1 (x) =

(−1)n 2 (2n − 1)! X sin 2kπ x (2π)2n−1 k2n−1 k=1

(n = 1 and 0 < x < 1; n ∈ N \ {1} and 0 5 x 5 1); E2n (x) =

(−1)n 4 (2n)! π 2n+1

∞ X k=0

sin (2k + 1)πx (2k + 1)2n+1

(n = 0 and 0 < x < 1; n ∈ N and 0 5 x 5 1);

(56) (57)

88

Zeta and q-Zeta Functions and Associated Series and Integrals ∞

E2n−1 (x) =

(−1)n 4 (2n − 1)! X cos (2k + 1)πx π 2n (2k + 1)2n

(n ∈ N; 0 5 x 5 1),

(58)

k=0

which, for x = 12 , yields E2n+1 = 0

(n ∈ N0 ),

(59)

where use is made of the relationship (48). The first few of the Euler numbers En are given below: E0 = 1, E2 = −1, E4 = 5, E6 = −61, E8 = 1385, E10 = −50521, . . . .

(60)

Relations Between Bernoulli and Euler Polynomials The following relationships between the Bernoulli and Euler polynomials follow easily from the definitions (1) and (40):     x  2n+1 x+1 Bn+1 − Bn+1 n+1 2 2  x o 2 n = Bn+1 (x) − 2n+1 Bn+1 (n ∈ N0 ), n+1 2

En (x) =

(61)

which, in view of (10), can also be written in the form:     2 x+1 n+1 2 Bn+1 − Bn+1 (x) En (1 − x) = (−1) n+1 2 n

(n ∈ N0 ).

(62)

Two additional formulas involving these polynomials are given below:  −1 X n−2     n n En−2 (x) = 2 2n−k − 1 Bn−k Bk (x) 2 k k=0   n X n Bn (x) = 2−n Bn−k Ek (2x) (n ∈ N0 ). k

(n ∈ N \ {1});

(63)

(64)

k=0

The Generalized Euler Polynomials and Numbers (α)

(α)

The generalized Euler polynomials En (x) and the generalized Euler numbers En are defined by the generating functions: 

2 z e +1

α exp(xz) =

∞ X n=0

En(α) (x)

zn n!

(|z| < π ; 1α := 1)

(65)

Introduction and Preliminaries

89

and 

2 ez e2z + 1

α =

∞ X

En(α)

n=0

zn n!



|z| <

 π α ; 1 := 1 2

(66)

for arbitrary (real or complex) parameter α. Clearly, we have En(1) (x) = En (x) and En(1) = En

(n ∈ N0 ).

(67)

It is easy to find from (65) and (66) that 2n En(α)

α  2

= En(α)

(n ∈ N0 ),

(68)

which, for α = 1, reduces to (48). Srivastava et al. [1101, p. 443, Eq. (4.12)] proved the following interesting addition theorem for the generalized Euler polynomials: En(α+λγ ) (x + γ y) =

  n X γ + n n (α−λk) (λk+λγ ) E (x − ky) En−k (ky + γ y) γ +k k k

(<(γ ) > 0).

k=0

(69) From the generating functions (20) and (65), it is easily seen that (0) n B(0) n (x) = En (x) = x

(n ∈ N0 ).

(70)

Recently, by making use of some fairly standard techniques based on series rearrangement, Srivastava and Pinte´ r [1105] derived each of the following elegant theorems (cf. [1105, p. 379, Theorem 1; p. 380, Theorem 2]). The following relationship (cf. [1105, p. 379, Theorem 1]): B(α) n (x + y) =

n   X n k=0

k

 k (α−1) (α) Bk (y) + Bk−1 (y) En−k (x)(α ∈ C; n ∈ N0 ) 2

(71)

holds true between the generalized Bernoulli polynomials and the classical Euler polynomials. The following relationship (cf. [1105, p. 380, Theorem 2]): En(α) (x + y) =

n X k=0

 h i 2 n (α−1) (α) Ek+1 (y) − Ek+1 (y) Bn−k (x)(α ∈ C; n ∈ N0 ), k+1 k (72)

holds true between the generalized Euler polynomials and the classical Bernoulli polynomials.

90

Zeta and q-Zeta Functions and Associated Series and Integrals

Upon setting α = 1 in (71), if we let y → 0 and make use of (70), we can deduce the aforementioned main relationship in Cheon’s work (cf. [254, p. 368, Theorem 3]): n   X n Bn (x) = Bk En−k (x) k

(n ∈ N0 ),

(73)

k=0 (k6=1)

just as it was accomplished by Srivastava and Pinte´ r [1105, p. 379].

Genocchi Polynomials and Numbers (α)

The Genocchi polynomials Gn (x) of (real or complex) order α are usually defined by means of the following generating function: 

2z ez + 1

α

· exz =

∞ X

G(α) n (x)

n=0

zn n!


|z| < π ; 1α := 1 ,

(74)

so that, obviously, the classical Genocchi polynomials Gn (x), given by Gn (x) := G(1) n (x)

(n ∈ N0 ) ,

(75)

are defined by the following generating function: ∞

X zn 2zexz = G (x) n ez + 1 n!

(|z| < π) .

(76)

n=0

For the classical Genocchi numbers Gn , we have (see also Problem 54) Gn := Gn (0) = G(1) n (0) .

(77) (α)

Finally, in light of the definition (74), we find for the Genocchi numbers Gn (real or complex) order α that 

2z ez + 1

α =

∞ X

G(α) n

n=0

zn n!


|z| < π ; 1α := 1 ,

of

(78)

so that, just as we observed above in Equation (77), Gn = G(1) n

(n ∈ N0 ).

Various properties and characteristics of the above-defined Genocchi polynomials and numbers, analogous to those that hold true for the Bernoulli and Euler polynomials and numbers, can be found in the widely-scattered literature, which is cited by (for example) Luo and Srivastava [791] (see also Section 1.8 below).

Introduction and Preliminaries

91

1.8 Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials and Numbers Apostol-Bernoulli Polynomials and Numbers Some interesting analogues of the classical Bernoulli polynomials and numbers were investigated by Apostol [58, p. 165, Eq. (3.1)] and (more recently) by Srivastava [1087, pp. 83–84]. We begin by recalling, here, Apostol’s definitions as follows: The Apostol-Bernoulli polynomials Bn (x; λ) are defined by means of the generating function (Apostol [58]; see also Srivastava [1087]): ∞

X zn zexz = B (x; λ) n λez − 1 n!

(1)

n=0

(|z| < 2π,

when λ = 1; |z| < |log λ| ,

when λ 6= 1)

with, of course, Bn (x) = Bn (x; 1)

and

Bn (λ) := Bn (0; λ)

(2)

where Bn (λ) denotes the so-called Apostol-Bernoulli numbers. Under the assumption λ 6= 1, Apostol [58] gave the main properties of Bn (a, λ), including, for example, the summation formula: Bn (a, λ) =

n   X n k=0

k

Bk (λ) an−k

(n ∈ N0 ),

(3)

from which it is seen that Bn (a, λ) are polynomials in a, satisfying the difference equation: λ Bn (a + 1, λ) − Bn (a, λ) = n an−1

(n ∈ N),

(4)

which yields the following special cases: λ B1 (1, λ) = 1 + B1 (λ)

(5)

λ Bn (1, λ) = Bn (λ)

(6)

and (n ∈ N\{1}).

92

Zeta and q-Zeta Functions and Associated Series and Integrals

Equations (5), (6) and (3), with a = 1, together, aid us in computing Bn (λ) recursively. Thus, we have 2λ 3λ(λ + 1) 1 , B2 (λ) = − , B3 (λ) = , 2 λ−1 (λ − 1) (λ − 1)3

4λ λ2 + 4λ + 1 5λ λ3 + 11λ2 + 11λ + 1 B4 (λ) = − , B5 (λ) = , 4 (λ − 1)5 (λ − 1)
6λ λ4 + 26λ3 + 66λ2 + 26λ + 1 B6 (λ) = − (λ − 1)6 B0 (λ) = 0,

B1 (λ) =

(7) and (in general) Bn (λ) =

n−1 nλ X (−1)k k! λk−1 (λ − 1)n−1−k S(n − 1, k), (λ − 1)n

(8)

k=1

in terms of the Stirling numbers of the second kind (see Section 1.6). We recall here several further properties of the functions Bn (a, λ) as follows: ∂p n! {Bn (a, λ)} = Bn−p (a, λ) (p = 0, 1, . . . , n); ∂ap (n − p)! n   X n Bn (a + b, λ) = Bk (a, λ) bn−k ; k

(9) (10)

k=0

Zb

Bn (t, λ) dt =

a m−1 X

kn =

k=0

Bn+1 (b, λ) − Bn+1 (a, λ) ; n+1

(11)

m λ−1 X Bn+1 (m, λ) − Bn+1 (λ) Bn+1 (k, λ) + , n+1 n+1

(12)

k=1

which obviously generalizes the familiar result 1.7(17). Motivated by the success of the generalizations in 1.7(20) and 1.7(66) of the classical Bernoulli polynomials and the classical Euler polynomials involving a real or complex parameter α, Luo and Srivastava [788, 789] introduced and investigated the (α) so-called Apostol-Bernoulli polynomials Bn (x; λ) of order α and the Apostol-Euler (α) polynomials En (x; λ) of order α, which are defined as follows: (α) The Apostol-Bernoulli polynomials Bn (x; λ) of (real or complex) order α are defined by means of the following generating function: 

z z λe − 1

α

|z| < 2π,

· exz =

∞ X n=0

Bn(α) (x; λ)

zn n!

when λ = 1; |z| < |log λ| ,

(13)
when λ 6= 1; 1α := 1 ,

Introduction and Preliminaries

93

with, of course, and Bn(α) (λ) := Bn(α) (0; λ) ,

(α) B(α) n (x) = Bn (x; 1)

(14)

(α)

where Bn (λ) denotes the so-called Apostol-Bernoulli numbers of order α. (α) The Apostol-Euler polynomials En (x; λ) of (real or complex) order α are defined by means of the following generating function (cf. Luo [781]):



2 λez + 1

α

· exz =

∞ X

En(α) (x; λ)

n=0

zn n!


|z| < |log(−λ)| ; 1α := 1 ,

(15)

with, of course, En(α) (x) = En(α) (x; 1)

and En(α) (λ) := En(α) (0; λ) ,

(16)

(α)

where En (λ) denotes the so-called Apostol-Euler numbers of order α. Luo and Srivastava [788, 789] presented a variety of properties and relations between other mathematical functions. Among them, we choose to record, here, the following results: (α) The Apostol-Euler polynomials En (x; λ) of order α is represented by En(α) (x; λ) = e−x log λ

∞ X k=0

(α)

En+k (x)

(log λ)k k!

(n ∈ N0 )

(17)

(α)

in series of the familiar Euler polynomials En (x) of order α. (`) The Aposto-Bernoulli polynomials Bn (x; λ) of order ` is represented by

Bn(`) (x; λ) = e−x log λ

  ∞  X n + k − ` n + k −1 (`) (log λ)k Bn+k (x) k k k!

(18)

k=0

(n, ` ∈ N0 ) (`)

in series of the familiar Bernoulli polynomials Bn (x) of order `. (`) An explicit representation of Bn (λ) is given:  X  n−`  n ` + k − 1 k! (−λ)k S (n − `, k) , ` k (λ − 1)k+`

Bn(`) (λ) = `!

k=0


n, ` ∈ N0 ; λ ∈ C \ {1} .

(19)

94

Zeta and q-Zeta Functions and Associated Series and Integrals (α)

An explicit formula for the Apostol-Bernoulli polynomials Bn (x; λ) involving the Stirling numbers of the second kind is also given as follows: Bn(`) (x; λ) = `!

n    X n k

`

k

k=`

x

n−k

 k−`  X ` + j − 1 j! (−λ)j S (k − `, j) j (λ − 1)j+`

(20)

j=0


n, ` ∈ N0 ; λ ∈ C \ {1} . The following relationship: Bn(α) (x; λ) =

n   X n k=0

k

(α−1)

Bn−k (λ) Bk (x; λ)

(n ∈ N0 )

(21)

(α)

holds true between the Apostol-Bernoulli polynomials Bn (x; λ) of order α and the (α−1) Apostol-Bernoulli numbers Bn (λ) of order α − 1. Let n ∈ N0 . Suppose also that α and λ are suitable (real or complex) parameters. Then, Bn(α) (x; λ) =

n   X n

(α)

B (λ) xn−k and Bn(0) (x; λ) = xn , k k k=0 n   (α) X α n−k n Ek (λ)  , x − En(α) (x; λ) = 2 k 2k

(22)

(23)

k=0

(α−1)

λBn(α) (x + 1; λ) − Bn(α) (x; λ) = nBn−1 (x; λ) , o ∂ n (α) (α) Bn (x; λ) = nBn−1 (x; λ) , ∂x Zb (α) (α) B (b; λ) − Bn+1 (a; λ) Bn(α) (x; λ) , dx = n+1 , n+1

(24) (25) (26)

a

Bn(α+β) (x + y; λ) =

n   X n

(α)

(β)

B (x; λ) Bn−k (y; λ) , k k k=0 n   X n (α) (β) (α+β) En (x + y; λ) = E (x; λ)En−k (y; λ), k k k=0 (−1)n (α)  −1  (α) Bn (α − x; λ) = Bn x; λ , λα n (−1) (α) En(α) (α − x; λ) = E (x; λ−1 ), λα n  (−1)n (α)  −1 Bn(α) (α + x; λ) = B −x; λ , n λα (α) nxBn−1 (x; λ) = (n − α)Bn(α) (x; λ) + αλBn(α+1) (x + 1; λ)

(27)

(28) (29) (30) (31) (32)

Introduction and Preliminaries

95

and  x  n  (α) (α) Bn(α+1) (x; λ) = 1 − Bn (x; λ) + n − 1 Bn−1 (x; λ) . α α

(33)

From the generating functions (13) and (15), it follows also that (see [788] and [781]) (α−1)

λBn(α) (x + 1; λ) − Bn(α) (x; λ) = nBn−1 (x; λ)

(34)

λEn(α) (x + 1; λ) + En(α) (x; λ) = 2En(α−1) (x; λ),

(35)

and

respectively. Now, since Bn(0) (x; λ) = En(0) (x; λ) = xn

(n ∈ N0 ),

(36)

upon setting β = 0 in the addition theorems (27) and (28), if we interchange x and y, we obtain n   X n (α) Bn(α) (x + y; λ) = B (y; λ)xn−k (37) k k k=0

and En(α) (x + y; λ) =

n   X n (α) E (y; λ)xn−k , k k

(38)

k=0

respectively. Next, by combining (34) and (37) (with x = 1 and y 7−→ x), we find that # " n+1  X n + 1 (α) 1 (α) λ Bk (x; λ) − Bn+1 (x; λ) (n ∈ N0 ), (39) Bn(α−1) (x; λ) = n+1 k k=0

which, in the special case when α = 1, yields the following expansion: " n+1  # X n + 1 1 n x = λ Bk (x; λ) − Bn+1 (x; λ) (n ∈ N0 ), n+1 k

(40)

k=0

in series of the Apostol-Bernoulli polynomials {Bn (x; λ)}∞ n=0 . In the special case of (40), when λ = 1, we obtain the following familiar expansion (cf., e.g., [795, p. 26]):  n  1 X n+1 Bk (x) x = n+1 k n

k=0

(n ∈ N0 )

(41)

96

Zeta and q-Zeta Functions and Associated Series and Integrals

in series of the classical Bernoulli polynomials {Bn (x)}∞ n=0 . In precisely the same manner, the addition theorem (38) in conjunction with (35) would lead us to the following companions of (39) and (40): " n   # X n (α) 1 λ E (x; λ) + En(α) (x; λ) En(α−1) (x; λ) = k k 2

(n ∈ N0 )

(42)

k=0

and " n   # X n 1 x = λ Ek (x; λ) + En (x; λ) 2 k n

(n ∈ N0 ).

(43)

k=0

In view of (36), this last expansion (43) in series of the Apostol-Euler polynomials {En (x; λ)}∞ n=0 is indeed an immediate consequence of (42), when α = 1. By using (13) (with α = 1) and (15) (with α = 1), we have ∞ X

Bn (x; λ2 )

n=0

t/2 2ext text tn = t/2 · t/2 = 2 t n! λ e − 1 λe − 1 λe + 1 ∞ X

∞

tn X −n tn · 2 En (2x; λ) n! n! n=0 n=0 " # ∞ n   X X n tn −n , = 2 Bn−k (λ)Ek (2x; λ) n! k =

n=0

2−n Bn (λ)

k=0

which yields the following relationship between the Apostol-Bernoulli and ApostolEuler polynomials: n   X n Bn−k (λ)Ek (2x; λ) k

(44)

n   X n = Bk (λ)En−k (x; λ). k

(45)

Bn (x; λ2 ) = 2−n

k=0

or, equivalently, n

2 Bn

x 2

;λ

2



k=0

By applying similar arguments, it is not difficult to get the following explicit representation for the Apostol-Euler polynomials En (x; λ) in terms of the Apostol-Bernoulli polynomials Bn (x; λ):     x 2n x+1 2 2 En−1 (x; λ) = λBn ; λ − Bn ; λ n 2 2

(46)

Introduction and Preliminaries

97

or, equivalently, En−1 (x; λ) =

x i 2h Bn (x; λ) − 2n Bn ; λ2 . n 2

(47)

In addition, from the relationships (46) (with x = 0) and (47) (with x = 0), we find that   1 2 (48) ; λ = 2−n Bn (λ) + n · 2−n−1 En−1 (0; λ). λBn 2 Thus, by substituting for En−1 (0; λ) from (47) (with x = 0) into (48), we obtain the above-asserted relationship:   (−1)n 1 Bn (λ) := Bn (0; λ) = Bn 1; λ λ    (49) λ 1 2 n 1 2 =2 (n ∈ N0 ), Bn (λ ) + Bn ;λ 2 2 2 that is,    1 2 Bn (λ) = 2n−1 Bn (λ2 ) + λBn ;λ 2

(n ∈ N0 ).

(50)

The following relationship: Bn(α) (x + y; λ) =

n   X n k=0

k

 k (α−1) (α) Bk (y; λ) + Bk−1 (y; λ) En−k (x; λ) 2

(51)

(α, λ ∈ C; n ∈ N0 ) holds true between the generalized Apostol-Bernoulli polynomials and the ApostolEuler polynomials. Luo [781] obtained the following general recursion formulas for the generalized (α) Apostol-Euler polynomials En (x; λ) and the generalized Apostol-Euler numbers (α) En (λ) (see [781, Equations (20) and (29)]): En(α) (x; λ) = 2α

 n   k  X n n−k X α + j − 1 j!(−λ) j S(k, j) x (λ + 1)j+α k j k=0

j=0

(α, λ ∈ C; n ∈ N0 )

(52)

and En(α) (λ) = (−1)n

 n   k  X n k+α n−k X α + j − 1 j!(−λ) j 2 α S(k, j) k j (λ + 1)j+α k=0

j=0

(α, λ ∈ C; n ∈ N0 ).

(53)

98

Zeta and q-Zeta Functions and Associated Series and Integrals

Luo and Srivastava [789] gave an addition formula for each of the generalized Apostol-Bernoulli and the generalized Apostol-Euler polynomials: Bn(α) (x + y; λ) =

n   X n−k   X x n (α) k! B (y; λ)S(n − j, k) k j j

(54)

n   X n−k   X x n (α) k! E (y; λ)S(n − j, k) k j j

(55)

k=0

j=0

and En(α) (x + y; λ) =

k=0

j=0

(α, λ ∈ C; n ∈ N0 ) hold true between the generalized Apostol-Bernoulli polynomials, the generalized Apostol-Euler polynomials and the Stirling numbers of the second kind. By setting λ = 1 in (54) and (55), it is easy to deduce the following interesting identities: B(α) n (x + y) =

n   X n−k   X x n (α) k! B (y)S(n − j, k) k j j

(α ∈ C; n ∈ N0 )

(56)

n   X n−k   X x n (α) k! E (y)S(n − j, k) k j j

(α ∈ C; n ∈ N0 )

(57)

k=0

j=0

and En(α) (x + y) =

k=0

j=0

for the generalized Bernoulli polynomials and the generalized Euler polynomials of order α.

Apostol-Genocchi Polynomials and Numbers Since the publication of the works by Luo and Srivastava (see [780], [781], [788] and [789]), many further investigations of the above-mentioned Apostol type polynomials have appeared in the literature. Boyadzhiev [162] gave some properties and representations of the Apostol-Bernoulli polynomials and the Eulerian polynomials. Garg et al. [467] studied the Apostol-Bernoulli polynomials of order α and obtained some new relations and formulas involving the Apostol type polynomials and the Hurwitz (or generalized) zeta function ζ (s, a) defined by 2.2(1) below. Luo (see [782] and [783]) obtained the Fourier expansions and integral representations for the ApostolBernoulli and the Apostol-Euler polynomials and gave the multiplication formulas for the Apostol-Bernoulli and the Apostol-Euler polynomials of order α. Pre´ vost [915] investigated the Apostol-Bernoulli and the Apostol-Euler polynomials by using the

Introduction and Preliminaries

99

Pade´ approximation methods. Wang et al. (see [1207] and [1208]) further developed some results of Luo and Srivastava [789] and obtained some formulas involving power sums of the Apostol type polynomials. Zhang and Yang [1258] gave several identities for the generalized Apostol-Bernoulli polynomials. Conversely, Cenkci and Can [227] gave a q-analogue of the Apostol-Bernoulli polynomials Bn (x; λ). Choi et al. [267] gave the q-extensions of the Apostol-Bernoulli polynomials of order α and the Apostol-Euler polynomials of order α (see also [268]). Hwang et al. [579] and Kim et al. [663] also gave q-extensions of Apostol’s type Euler polynomials. On the subject of the Genocchi polynomials Gn (x) and their various extensions, a remarkably large number of investigations have appeared in the literature (see, e.g., [229], [268], [566], [567], [568], [604], [655], [656], [657], [720], [721], [722], [773], [784], [785], [786], [886] and [1257]; see also the references cited in each of these works). Moreover, Luo (see [784] and [786]) introduced and investigated the Apostol-Genocchi polynomials of (real or complex) order α, which are defined as follows. The Apostol-Genocchi polynomials Gn(α) (x; λ) (λ ∈ C)

of (real or complex) order α are defined by means of the following generating function: 

2z z λe + 1

α

· exz =

∞ X

Gn(α) (x; λ)

n=0

zn n!

|z| < |log(−λ)| ; 1α := 1




(58)

with, of course, (α) G(α) n (x) = Gn (x; 1) ,

Gn (x; λ) := Gn(1) (x; λ)

Gn(α) (λ) := Gn(α) (0; λ) ,

and

Gn (λ) := Gn(1) (λ) ,

(59) (60)

(α)

where Gn (λ), Gn (λ) and Gn (x; λ) denote the so-called Apostol-Genocchi numbers, the Apostol-Genocchi numbers of order α and the Apostol-Genocchi polynomials, respectively.

Important Remarks and Observations The constraints on |z|, which we have used in the definitions (1), (13), (15) and (58) above, are meant to ensure that the generating functions in (1), (13), (15) and (58) are analytic throughout the prescribed open disks in the complex z-plane (centred at the origin z = 0) to have the corresponding convergent Taylor-Maclaurin series expansions (about the origin z = 0) occurring on their right-hand sides (each with a positive radius of convergence). Moreover, throughout this investigation, log z is tacitly assumed to denote the principal branch of the many-valued function log z

100

Zeta and q-Zeta Functions and Associated Series and Integrals



with the imaginary part I log z constrained by −π < I log z 5 π . More importantly, throughout this presentation, wherever | log λ| and | log(−λ)| appear as the radii of the open disks in the complex z-plane (centred at the origin z = 0) in which the defining generating functions are analytic, it is tacitly assumed that the obviously exceptional cases when λ = 1 and λ = −1, respectively, are to be treated separately. Naturally, therefore, the corresponding constraints on |z| in the earlier investigations (see, e.g., [781], [788], [789] and [1087]) and elsewhere in the literature dealing with one or the other of these Apostol type polynomials (see also [58]) should also be modified accordingly.

Generalizations and Unified Presentations of the Apostol Type Polynomials The mutual relationships among the families of the generalized Apostol-Bernoulli polynomials, the generalized Apostol-Euler polynomials and the generalized ApostolGenocchi polynomials, which are asserted by Problems 70, 71 and 72 below, can be appropriately applied with a view to translating various formulas involving one family of these generalized polynomials into the corresponding results involving each of the other two families of these generalized polynomials. Nevertheless, we find it useful to investigate properties and results involving these three families of generalized Apostol type polynomials in a unified manner. In fact, the following interesting unification (and generalization) of the generating functions of the three families of Apostol type polynomials was recently investigated, rather systematically, by Ozden et al. (cf. [886, p. 2779, Equation (1.1)]): zn 21−κ zκ exz X = Y (x; κ, a, b) n,β n! β b ez − ab n=0    β |z| < 2π when β = a; |z| < b log a  1α := 1; κ, β ∈ C; a, b ∈ C \ {0} , ∞

(61) when β 6= a;

where we have not only suitably relaxed the constraints on the parameters κ, a and b, but we have also strictly followed the above remarks and observations regarding the open disk in the complex z-plane (centred at the origin z = 0) within which the generating function in (61) is analytic to have the corresponding convergent TaylorMaclaurin series expansion (about the origin z = 0) occurring on the right-hand side (with a positive radius of convergence). Here, in conclusion of our present section, we first define the following unification (and generalization) of the generating functions of the above-mentioned three families of the generalized Apostol type polynomials.

Introduction and Preliminaries

101

Definition 1.1 The generalized Apostol type polynomials Fn(α) (x; λ; µ; ν)

(α, λ, µ, ν ∈ C)

of (real or complex) order α are defined by means of the following generating function: 

2µ zν λez + 1

α

· exz =

∞ X

Fn(α) (x; λ; µ; ν)

n=0

zn n!


|z| < |log(−λ)| ; 1α := 1 ,

(62)

so that, by comparing Definition 1 with the corresponding definitions given above, we have Bn(α) (x; λ) = (−1)α Fn(α) (x; −λ; 0; 1) ,

(63)

En(α) (x; λ) = Fn(α) (x; λ; 1; 0)

(64)

Gn(α) (x; λ) = Fn(α) (x; λ; 1; 1) .

(65)

and

Furthermore, if we compare the generating functions (61) and (62), we have Yn,β (x; κ, a, b) = −

  b  β 1 (1) F x; − ; 1 − κ; κ . a ab n

(66)

We, thus, see from the relationships (63), (64), (65) and (66), that the generating (α) function of Fn (x; λ; µ; ν) in (62) includes, as its special cases, not only the generating function of the polynomials Yn,β (x; κ, a, b) in (61) and the generating functions (α) (α) of all three of the generalized Apostol type polynomials Bn (x; λ), En (x; λ) and (α) (α) (α) Gn (x; λ), but also the generating functions of their special cases Bn (x), En (x) (α) and Gn (x). The various interesting properties and results involving the new unified family of (α) generalized Apostol type polynomials Fn (x; λ; µ; ν), given by Definition 1 above, can also be derived in a manner analogous to that of our investigation in this presentation. The following natural generalization and unification of the Apostol-Bernoulli (α) polynomials Bn (x; λ) of order α, as well as the generalized Bernoulli numbers Bn (a, b) studied by Guo and Qi [519] and the generalized Bernoulli polynomials Bn (x; a, b) studied by Luo et al. [787], was introduced and investigated recently by Srivastava et al. [1095].

102

Zeta and q-Zeta Functions and Associated Series and Integrals

Definition 1.2 (cf. [1095, p. 254, Equation (20)]). The generalized Apostol-Bernoulli (α) type polynomials Bn (x; λ; a, b, c) of order α ∈ C are defined by the following generating function:  α ∞ X z zn xz (α) λ; a, b, c) (67) · c = B (x; n λbz − az n! n=0   log λ |z| <   ; a ∈ C \ {0}; b, c ∈ R+ ; a 6= b; 1α := 1 . log ba In a sequel to the work by Srivastava et al. [1095], a similar generalization of each of the families of Euler and Genocchi polynomials were introduced and investigated (see, for details, [1096, Section 4]). Definition 1.3 (cf. [1096, Section 2]). The generalized Apostol-Euler type polynomi(α) als En (x; λ; a, b, c) of order α ∈ C are defined by the following generating function: α  ∞ X zn 2 xz · c = (68) E(α) n (x; λ; a, b, c) z z λb + a n! n=0   log(−λ) |z| <   ; a ∈ C \ {0}; b, c ∈ R+ ; a 6= b; 1α := 1 . log ab Definition 1.4 (cf. [1096, Section 4]). The generalized Apostol-Genocchi type polyno(α) mials Gn (x; λ; a, b, c) of order α ∈ C are defined by the following generating function:  α ∞ X 2z zn xz (α) · c = G λ; a, b, c) (69) (x; n λbz + az n! n=0   log(−λ) |z| <   ; a ∈ C \ {0}; b, c ∈ R+ ; a 6= b; 1α := 1 . log ab Remark 1 In their special case when a=1

and b = c = e, (α)

the generalized Apostol-Bernoulli type polynomials Bn (x; λ; a, b, c) defined by (α) (67), the generalized Apostol-Euler type polynomials En (x; λ; a, b, c) defined by (α) (68) and the generalized Apostol-Genocchi type polynomials Gn (x; λ; a, b, c) (α) defined by (69) would reduce at once to the Apostol-Bernoulli polynomials Bn (x; λ), (α) the Apostol-Euler polynomials En (x; λ) and the Apostol-Genocchi polynomials (α) Gn (x; λ), respectively.

Introduction and Preliminaries

103

Since the parameter λ ∈ C, by comparing Definitions 2, 3 and 4 above, we can easily deduce the following potentially useful lemma (see also Lemmas 1, 2 and 3). Lemma 1.7 The families of the generalized Apostol-Bernoulli type polynomials B(l) n (x; λ; a, b, c)

(l ∈ N0 )

and the generalized Apostol-Euler type polynomials E(l) n (x; λ; a, b, c)

(l ∈ N0 )

are related by   n! 1 l (l) E (x; −λ; a, b, c) B(l) λ; a, b, c) = − (x; n 2 (n − l)! n−l

(n, l ∈ N0 ; n = l) (70)

or, equivalently, by l E(l) n (x; λ; a, b, c) = (−2)

n! (l) B (x; −λ; a, b, c) (n + l)! n+l

(n, l ∈ N0 ).

(71)

Furthermore, the families of the generalized Apostol-Bernoulli type polynomials B(l) n (x; λ; a, b, c)

(l ∈ N0 )

and the generalized Apostol-Euler type polynomials E(l) n (x; λ; a, b, c)

(l ∈ N0 )

are related to the generalized Apostol-Genocchi type polynomials G(l) n (x; λ; a, b, c)

(l ∈ N0 )

as follows: Gn(α) (x; λ; a, b, c) = (−2)α B(α) n (x; −λ; a, b, c)

α ∈ C; 1α := 1




(72)

and (l)

l G(l) n (x; λ; a, b, c) = (−1) (−n)l En−l (x; λ; a, b, c) =


n, l ∈ N0 ; n = l; λ ∈ C .

n! (l) E (x; λ; a, b, c) (n − l)! n−l

104

Zeta and q-Zeta Functions and Associated Series and Integrals

The inter-relationships asserted by the above Lemma do aid in translating the various properties and results involving any one of these three families of generalized Apostol type polynomials in terms of the corresponding properties and results involving the other two families. Nonetheless, it would occasionally seem more appropriately convenient to investigate these three families in a unified manner by means of Definition 5 below. Definition 1.5 A unification of the generalized Apostol-Bernoulli type polynomials B(α) n (x; λ; a, b, c) , the generalized Apostol-Euler type polynomials E(α) n (x; λ; a, b, c) and the generalized Apostol-Genocchi type polynomials Gn(α) (x; λ; a, b, c) of order α ∈ C is defined by the following generating function:  ∞ zn 2µ zν α xz X (α) · c = Z λ; a, b, c; µ; ν) (73) (x; n λbz + az n! n=0   log(−λ) |z| <   ; a ∈ C \ {0}; b, c ∈ R+ ; a 6= b; α, λ, µ, ν ∈ C; 1α := 1 , log ba



so that, by comparing Definition 5 with Definitions 1 through 4, we have Fn(α) (x; λ; µ; ν) = Zn(α) (x; λ; 1, e, e; µ; ν) , α (α) B(α) n (x; λ; a, b, c) = (−1) Zn (x; −λ; a, b, c; 0; 1) , En(α) (x; λ; a, b, c) = Zn(α) (x; λ; a, b, c; 1; 0)

(74) (75) (76)

and Gn(α) (x; λ; a, b, c) = Zn(α) (x; −λ; a, b, c; 1; 1) .

(77)

Thus, clearly, Definitions 1 and 5 above provide us with remarkably powerful and extensive generalizations of the various families of the Apostol type polynomials and Apostol type numbers. Properties and results involving these generalizations deserve to be investigated further (see also [791], [886], [1095] and [1096]; see also [1091] and Problem 73 onwards of this chapter).

Introduction and Preliminaries

105

1.9 Inequalities for the Gamma Function and the Double Gamma Function The Gamma Function and Its Relatives Recently, many research articles were published providing inequalities for the Gamma function and its relatives. We refer to Gautschi’s survey paper [476] and the comprehensive bibliography complied by Sa´ ndor [1003]. We begin by proving that log 0 is convex on (0, ∞) (cf. Theorem 1.1):     1 1 x y 1/p 1/q 1 < p < ∞; + = 1 . + 5 [0(x)] [0(y)] (1) 0 p q p q Indeed, if 1 < p < ∞ and 1/p + 1/q = 1, then we have 

x y 0 + p q

Z∞

 =

x

tp

+ qy −1 −t

e dt

0

Z∞ =

t

x−1 p

t

y−1 q

e−(1/p+1/q)t dt

0

∞ 1/q 1/p  ∞ q Z  Z  y−1 p x−1 5 t p e−t/p dt  t q e−t/q dt 0

0

∞ 1/p  ∞ 1/q Z Z =  tx−1 e−t dt  ty−1 e−t dt 0

0

= [0(x)]1/p [0(y)]1/q , where we made use of the well-known Ho¨ lder’s inequality: For 1 5 p 5 ∞, the spaces Lp (µ) are Banach spaces and, if f ∈ Lp (µ) and g ∈ Lq (µ) (with 1/p + 1/q = 1), then fg ∈ L1 (µ) and Z

|f (µ)g(µ)| dµ 5

Z

1/p Z 1/q q | f (µ)| dµ |g(µ)| dµ . p

(2)

We recall, here (see, e.g., [115, 406, 932, 939, 1184, 1226]), that a function f is said to be strictly completely monotonic on an interval I ⊂ R, if (−1)n f (n) (x) > 0

(x ∈ I; n ∈ N0 ).

(3)

If (−1)n f (n) (x) = 0 for all x ∈ I and n ∈ N0 , then f is called completely monotonic on I. We recall also (see, e.g., [76, 935, 939, 943, 944, 948, 950]) that a positive function

106

Zeta and q-Zeta Functions and Associated Series and Integrals

f is said to be logarithmically completely monotonic on an interval I ⊂ R, if (−1)n {ln f (x)}(n) = 0

(x ∈ I; n ∈ N).

(4)

For convenience, C [I] and L[I] denote, respectively, the sets of completely monotonic functions and the logarithmically completely monotonic functions on an interval I ⊂ R. It is known (see [115, 926, 935, 939, 943, 944]) that L[I] ⊂ C [I]. Alzer [22] remarked that completely monotonic functions play a dominant roˆ le in areas, such as numerical analysis, probability theory and physics. The concept of complete and logarithmically complete monotonicities has also played an important roˆ le to prove some inequalities involving the gamma function. Here, we investigate some known inequalities involving the gamma function, by, mainly, focusing on C [I] and L[I]. Wendel [1223] proved the following double inequality: 

x x+a

1−a 5

0(x + a) 51 xa 0(x)

(0 < a < 1; x > 0),

(5)

which can be rewritten as follows: x1−a 5

0(x + 1) 5 (x + a)1−a 0(x + a)

(0 < a < 1; x > 0),

(6)

to establish the well-known asymptotic relation (see 1.1(57)): lim

x→∞

0(x + a) = 1 (a, x ∈ R), xa 0(x)

(7)

by using the Ho¨ lder’s inequality (2). Komatu [689] proved the inequality 1 2 < ex √ x + x2 + 2

Z∞

2

e−t dt 5

x

1 √ x + x2 + 1

(0 5 x < ∞).

(8)

Pollak [905] has improved the upper bound in (8), by showing that x2

Z∞

e

x

2

e−t dt 5

1 p . x + x2 + 4/π

(9)

Gautschi [470] proved more general inequalities than those in (8) and (9):    Z∞ i
1/p 1h p 1 p xp x +2 −x < e −x e−t dt 5 cp x p + 2 cp

(0 5 x < ∞),

x

(10)

Introduction and Preliminaries

107

where    1 p/(p−1) cp := 0 1 + p

(p ∈ N \ {1}).

For p = 2, the right-hand inequality of (10) reduces to (9), whereas the left-hand inequality reduces to the corresponding inequality in (8). The integral in (10) for p = 3 occurs in heat transfer problems [1242], for p = 4 in the study of electrical discharge through gases [1009]. An application of (10) for general p is given in [840]. Gautschi [470] also derived inequalities for the following Gamma-function ratio: 0(n + 1) , 0(n + s) which are given by n1−s 5

0(n + 1) 5 exp [(1 − s) ψ(n + 1)] 0(n + s)

n1−s 5

0(n + 1) 5 (n + 1)1−s 0(n + s)

(0 5 s 5 1; n ∈ N)

(11)

and (0 5 s 5 1; n ∈ N),

(12)

where ψ denotes the Psi-function defined in 1.3(1). The inequalities (11) have attracted remarkable interest, and several intriguing papers on the subject were subsequently published by, for example, Erber [420], Kec˘ kic´ and Vasic´ [642], Laforgia [724] and Zimering [1266], providing new bounds for the following Gamma-function ratio: 0(n + 1) . 0(n + s) However, it is observed that the upper bound in (12) is not better and the range in (12) is not broader than the corresponding ones in (6). Kershaw [645] gave proofs of the following closer bounds than (11):    h  i 0(x + 1) s+1 1/2 exp (1 − s) ψ x + s < exp (1 − s) ψ x + < 0(x + s) 2

(13)

"   #1−s  s 1−s 0(x + 1) 1 1 1/2 x+ < < x− + s+ , 2 0(x + s) 2 4

(14)

and

each being valid for x > 0 and 0 < s < 1.

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Zeta and q-Zeta Functions and Associated Series and Integrals

Bustoz and Ismail [196] established a remarkably more general result. They showed that the two functions    0(x + s) s+1 f1 (x) = exp (1 − s) ψ x + (0 < s < 1) (15) 0(x + 1) 2 and f2 (x) =

s s−1 0(x + 1)  x+ 0(x + s) 2

(0 < s < 1)

(16)

are strictly completely monotonic on (0, ∞). Since lim f1 (x) = lim f2 (x) = 1,

x→∞

x→∞

the inequalities (13) and (14) are immediate consequences of the fact that f1 and f2 are strictly decreasing on (0, ∞). Alzer [22] refined one of the Kec˘ kic´ -Vasic´ ’s inequalities in [642] and also gave the following interesting result: For every s ∈ (0, 1), the function x 7−→ fα (x, s) =

0(x + s) (x + 1)x+1/2 0(x + 1) (x + s)x+s−1/2    1  0 0 · exp s − 1 + (α > 0) ψ (x + 1 + α) − ψ (x + s + α) 12

(17)

is strictly completely monotonic on (0, ∞), if and only if α = 12 . Furthermore, for every s ∈ (0, 1), the function x 7−→

1 fβ (x, s)

(β = 0)

is strictly completely monotonic on (0, ∞), if and only if β = 0. Gurland [527] obtained the following inequality [0(λ + α)]2 λ < 0(λ) 0(λ + 2α) α 2 + λ

(α, λ ∈ R; α + λ > 0; λ > 0; α 6= 0; α 6= 1),

(18)

by making a novel use of the Crame´ r-Rao lower bound for the variance of an unbiased estimator (see Crame´ r [343]). Indeed, we consider the density function given by f (x) =

 x 1 exp − x λ−1 θ λ 0(λ) θ

From the fact that the expression: 0(λ) xα 0(α + λ)

(x > 0; λ > 0; θ > 0).

(19)

Introduction and Preliminaries

109

is an unbiased estimator of θ α , we have the following Crame´ r-Rao bound for the variance:  V

 1 0(λ) α x =  2 , 0(α + λ) E ∂θ∂ α {log f (x)}

which yields the inequality (18). A special case of (18) with λ = n/2 and α = 1/2 is seen to be reduced to Gurland’s formula [526]: 4n + 3 (2n + 1)2



(2n)!! (2n − 1)!!

2

<π <

4 4n + 1



(2n)!! (2n − 1)!!

2

(n ∈ N),

(20)

where (2n)!! := 2 · 4 · · · (2n − 2)(2n) and (2n − 1)!! := 1 · 3 · 5· · ·(2n − 3)(2n − 1). The inequality (20) is a sharper inequality for approximating π than that given by Wallis: 2 2n + 1



(2n)!! (2n − 1)!!

2

<π <

1 n



(2n)!! (2n − 1)!!

2

(n ∈ N).

(21)

Gokhale [492] obtained a similar inequality to (18) as follows: 0(λ) 0(λ + 2α) α 2 (λ − 2) > 1+ 2 [0(λ + α)] (λ + α − 1)2

(α, λ ∈ R; λ + 2α > 0,

(22)

λ > 2, α 6= 0, α 6= −1). Gurland’s inequality (18) can be written in the form: 0(λ) 0(λ + 2α) α2 > 1 + . λ [0(λ + α)]2

(23)

We note that the inequality (22) is seen to be stronger than that in (23) for a certain range of α. Indeed, when α 2 (λ − 2) α2 > , that is, λ (λ + α − 1)2 p p − (λ − 1) − λ(λ − 2) < α < −(λ − 1) + λ(λ − 2).

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Zeta and q-Zeta Functions and Associated Series and Integrals

By employing the multivariate generalization of (19), that is, the Wishart distribution, Olkin [878] obtained p−1 Y j=0


2 λ2 p2 − 1 + λ p4 [0(λ + α − j/2)]2 ≤
0(λ − j/2) 0(λ + 2α − j/2) λ2 p2 − 1 2 + λ p4 + α 2   p−1 α, λ ∈ R; α + λ > ; λ > 0; p > 0 . 2

(24)

The special case of (24) with p = 1 reduces to (18). Selliah [1018], by employing the multiparameter version of the Crame´ r-Rao lower bound, using the information matrix for the same problem, obtained the following inequality: p−1 Y j=0

λ [0(λ + α − j/2)]2 ≤ 0(λ − j/2) 0(λ + 2α − j/2) λ + p α 2   p−1 α, λ ∈ R; α + λ > ; λ > 0; p > 0 , 2

(25)

which is seen to be sharper than that given in (24). For x ∈ (−α, ∞), define the function zs,t (x) by    0(x + t) 1/(t−s)   −x   0(x + s) zs,t (x) :=      ψ(x+s) e − x (s = t),

(s 6= t) (26)

+ where s, t ∈ R+ 0 , R0 being the set of nonnegative real numbers, and α = min{s, t}. A monotonicity and convexity of zs,t (x) was proved (see [241, 406, 929, 945]) so that the function zs,t (x) is either convex and decreasing for |t − s| < 1 or concave and increasing for |t − s| > 1. From this fact, the best bounds in the Kershaw’s double inequality (14) could be deduced. Qi [930, 931] further generalized this result. For x ∈ (−ρ, ∞), define the function Ha,b,c (x) by

Ha,b,c (x) := (x + c)b−a

0(x + a) 0(x + b)

(a, b, c ∈ R; ρ = min{a, b, c}).

(27)

Very recently, Qi [932, Theorem 1] proved the following results: Ha,b,c (x) ∈ L[(−ρ, ∞)]

(a, b, c) ∈ D1 (a, b, c)




(28)

and 

Ha,b,c (x)

−1

∈ L[(−ρ, ∞)]


(a, b, c) ∈ D2 (a, b, c) ,

(29)

Introduction and Preliminaries

111

where, for convenience,    1 1 ∪ (a, b, c) | a > b = c + D1 (a, b, c) := (a, b, c) | a + b = 1, c 5 b < c + 2 2   ∪ (a, b, c) | 2a + 1 5 a + b 5 1, a < c ∪ (a, b, c) | b − 1 5 a < b 5 c 

\ {(a, b, c) | a = c + 1, b = c}

and     1 1 D2 (a, b, c) := (a, b, c) | a + b = 1, c 5 a < c + ∪ (a, b, c) | b > a = c + 2 2   ∪ (a, b, c) | b < a 5 c ∪ (a, b, c) | b + 1 5 a, c 5 a 5 c + 1  ∪ (a, b, c) | b + c + 1 5 a + b 5 1 \ {(a, b, c) | a = c + 1, b = c} \ {(a, b, c) | b = c + 1, a = c} . Qi [932, Theorem 2] made use of (28), (29) and 1.1(37) to prove the following inequalities: (x + c)a−b <

0(x + a) 0(x + b)

(x ∈ (−ρ, ∞); (a, b, c) ∈ D1 (a, b, c))

(30)

and 0(x + a) 0(δ + a) 5 0(x + b) 0(δ + b)



x+c δ+c

a−b

(x ∈ [δ, ∞); (a, b, c) ∈ D1 (a, b, c)) , (31)

where a, b, c ∈ R, ρ = min{a, b, c}, and δ is a constant greater than −ρ. If (a, b, c) ∈ D2 (a, b, c), then inequalities in (30) and (31) are reversed, respectively, in (−ρ, ∞) and [δ, ∞). Qi [932] then observed the following facts: setting a = 1 and 0 < b < 1 in (30) reveals that (x + b)1−b <

0(x + 1) 0(x + b)

(0 < b < 1; x ∈ (−b, ∞))

(32)

holds true. It is obvious that the inequality in (32) not only refines the lower bound, but also extends the range of the left-hand side of the inequality in (14). Taking a = 1, 0 < b < 1 and δ = 1 in (31) shows that 0(x + 1) 1 5 0(x + b) 0(1 + b)



x+b 1+b

1−b

(0 < b < 1; x ∈ [1, ∞))

holds true. A usual argument shows that, if

(33)

112

Zeta and q-Zeta Functions and Associated Series and Integrals



1 2

x=

 q √ − b + 14 (1 + b) 1−b 0(1 + b) + 1 := λ(b), √ (1 + b) 1−b 0(1 + b) − 1

then the inequality (33) would be better than the right-hand side of (14). It is easy to find that

lim λ(b) and

b→0+

lim λ(b) =

b→1−

 √  e + 1 − 5 eγ 2 eγ − e

∼ = 0.6123686 · · · < 1,

where γ denotes the Euler-Mascheroni constant defined by 1.1(3). This implies that the inequality in (33) refines the right-hand side of (14), if b is close enough to 1 and that the upper bound in (33) is better than the one in (14), if x is sufficiently large.

The Double Gamma Function In a striking contrast to an abundant literature on the inequalities for the Gamma function and its relatives, there has been a few known results for the inequalities for the double Gamma function (see, e.g., [446], [105] and [244]). Presumably, Batir [105] initiated the study of the following class of inequalities for the double Gamma function 02 (x + 1) = 1/G(x + 1): x

x

x

x2

(0(x)) 2 xx (2π) 2 e− 2 − 2 < G(x + 1) <



0(x) 0(x/2)

x

x

x

x2

(8π) 2 e− 2 − 2


x ∈ R+ ; (34)

x x2 x2 2− 2 + 2

ψ(x) < G(x + 1) < (2π) 2 e   x x x2 x2 (2π) 2 (0(x + 1))x exp − − − ψ(α(x)) 2 2 2   x x x2 x2 < G(x + 1) < (2π) 2 (0(x + 1))x exp − − − ψ(β(x)) 2 2 2 x

(2π) 2 e

ψ(x/2)

2 2 − x2 + x2

x


x ∈ R+ ;

(35) (36)
x ∈ R+ ,

where R+ denotes the set of positive real numbers, α(x) =

x 3

and β(x) =

x2 1 . 2 (x + 1) log(x + 1) − x

Problems 1. Show that each form of the Gamma function, defined by 1.1(1), 1.1(2) and 1.1(7), is analytic in its given domain. (Whittaker and Watson [1225, Chapter 12]; Rainville [959, pp. 15–18])

Introduction and Preliminaries

113

2. Use the Wielandt’s theorem [Theorem 1.3] to derive each of the following results: l

l

l

l

the Gauss product 1.1(4) from the Euler integral 1.1(1), Gauss’s multiplication formula 1.1(31), the representation of the Beta function by Gamma functions 1.1(42), Stirling’s formula 1.1(33).

(Remmert [973]) 3. Prove the following series representations for the Euler-Mascheroni constant γ : γ = 1+

  ∞ X (−1)n log(n − 1) n log 2 n=3

and γ=

  ∞ X (−1)n log n , n log 2 n=1

where the bracket denotes the greatest integer function. (Gerst [480]; Sandham [1001]) 4. Deduce the explicit Weierstrass canonical product form 1.3(3) for the double Gamma function 02 , by using the three properties associated with Barnes’s definition of 02 . (Barnes [94, pp. 265–269]; Whittaker and Watson [1225, p. 264]) 5. Prove the following generalization of Gauss’s multiplication formula 1.1(51) for the Gamma function: m−1 Y `=0



k` 0 kz + m



= (2π )

·

k−1 Y n=0

1 2 (m−k)

 mkz+ 1 (mk−m−k) 2 k m

 mn  0 mz + k

(k, m ∈ N).

(Choi and Quine [278, p. 131]; Magnus et al. [795, p. 3]) 6. For the Euler-Mascheroni constant γ , the sequence {sn }∞ n=0 , defined by   1 1 sn := 1 + + · · · + − log n 2 n−1

(n ∈ N \ {1}),

satisfies the asymptotic property:   sn = γ + O n−1

(n → ∞).

Show that γ − sn (n ∈ N \ {1}) can be expressed, as follows, as an infinite sum with rational terms: γ − sn =

∞ 1 X tm+2 m+n
n m m=0

(n ∈ N \ {1}),

114

Zeta and q-Zeta Functions and Associated Series and Integrals

where 1 tm+2 = − (m + 1)!

Z1

(0 − x)(1 − x) · · · (m − x) dx

(m ∈ N0 ) .

0

(Elsner [416, p. 1537]; Jolley [613, pp. 14–15]). 7. Prove the von Staudt-Clausen theorem (Theorem 6.1): B2n = In −

X 1 p

(n ∈ N),

p−1|2n

where Bn is the Bernoulli number, In is an integer and the sum is taken over all primes p such that p − 1 divides 2n. (Carath´eodory [210, pp. 281–284]; Apostol [65, pp. 274–275]) 8. Prove the following rapidly converging series expansion for ψ(z):   X n ∞ 1 1 X 1 1 log (z + n)2 + (z + n) + − − (k − n)Ak , 2 3 z+k 9

ψ(z) =

k=0

k=n+1

where Z1 Ak :=

  1 1 −1 (z + k + t)−2 (z + k + t)4 − (z + k + t)2 + dt. 3 9

0

(Shafer and Lossers [1020]) 9. Let Sn denote the area of the sphere of radius 1 in Rn . Prove that !n R∞ −t2 e dt √
n 2 π −∞
= (n ∈ N). Sn = ∞ R 0 2n 2 e−r rn−1 dr 0

(Campbell [208, pp. 126–128]) 10. Let Z In :=

Z ···

f

n X

!

tk t1α1 −1 · · · tnαn −1 dt1 · · · dtn

k=1

αk > 0; tk = 0; k = 1, · · · , n;

n X

! tk 5 1 ,

k=1

where f is a continuous function. Prove that 0(α1 ) · · · 0(αn ) In = 0(α1 + · · · + αn )

Z1

f (τ ) τ α1 +···+αn −1 dτ.

0

(Whittaker and Watson [1225, p. 258]; Campbell [208, pp. 128–133])

Introduction and Preliminaries

115

11. Prove that Zπ

π 0(λ) e− 2 πµ     2λ−1 0 λ+iµ+1 0 λ−iµ+1 2 2 1

(sin θ )λ−1 e−µθ dθ =

0

(<(λ) > 0; µ ∈ C).

(Nielsen [862, p. 159]) 12. Show that Z∞ −∞

e−2πizt 1 dt = cosh (π t) cosh (π z)

  1 1 − < =(z) < 2 2

(cf. Equation 1.1(46)) 13. Making use of the Pochhammer symbol defined by 1.1(5), show that the Vandermonde convolution theorem 1.4(9) can be rewritten in the form: n n X (α + β)n X (α)n−k (β)k (α)k (β)n−k = = k! (n − k)! n! (n − k)! k!

(α, β ∈ C; n ∈ N0 ) .

k=0

k=0

14. Show that Euler’s transformation 1.5(21) is equivalent to the Pfaff-Saalschu¨ tz theorem: " 3 F2

a, b, −n; c, a + b − c − n + 1;

# 1 =

(c − a)n (c − b)n (c)n (c − a − b)n


c 6∈ Z− 0 ; n ∈ N0 .

15. Deduce Gauss’s summation theorem 1.5(7) as a limit case of the Pfaff-Saalschu¨ tz theorem (see Problem 13 above). [Hint: Let n → ∞, and apply the asymptotic expansion 1.1(37).] 16. For the Bernoulli polynomials Bn (x) generated by 1.7(1), show that xn =

n X k=0

  n+1 1 Bk (x) k n+1

(n ∈ N0 ) .

Hence (or otherwise), deduce the following representation for the Laguerre polynomials (α) Ln (x) defined by 1.5(72): Ln(α) (x) =

  n X (−1)k n + α Bk (x) · 2 F2 (−n + k, 1; α + k + 1, 2 ; 1). k! n−k k=0

(cf. Magnus et al. [795, p. 26]; see also Popov [907]) 17. Derive the following asymptotic expansion for the Psi (or Digamma) function: ψ(z) = log z −

  1 1 1 1 − + − + O z−8 2z 12z2 120z4 252z6 (|z| → ∞; | arg(z)| 5 π −  (0 <  < π)).

116

Zeta and q-Zeta Functions and Associated Series and Integrals

18. For a, b ∈ N, let (a, b) and [a, b] denote the greatest common divisor and least common multiple, respectively, of a and b, and {x} = x − [x] denote the fractional part of x. Then, show that, if <(s) > 12 , Z1

ζ (1 − s, {ax}) ζ (1 − s, {bx}) dx =

0

2 {0(s)}2 ζ (2s) (2π )2s



(a, b) [a, b]

s

,

where ζ (s, a) is the Hurwitz (or generalized) Zeta function defined by 2.2(1). Show also that Z1

Bn ({ax}) Bn ({bx}) dx = (−1)n−1

0

B2n (2n)!



(a, b) [a, b]

n

(n ∈ N).

(cf. Mordell [844, p. 372]; see also Mikol´as [827]) 19. Prove that √ 1/ Z 8 0

i 1 h 2 t2 log t 2
1 dt = 96 2π + 30(log 2) − 39 log 2 − 9 . t2 + 1 2 (Knuth [679, p. 138])

20. Prove that Z∞ 0

cosh x · log x dx cosh (2x) − cos (2π a)      1 1   0 a + 2 2 aπ π 1 log      + log 2π cot = 2 sin (π a) 2 2 0 1a

(0 < a < 1).

2

(Williams and Zhang [1232, p. 44]) 21. Prove the following inequality: cosec2 x −

m X 1 1 < < cosec2 x 2m + 1 (x − kπ )2 k=−m

1 (m ∈ N; 0 < |x| 5 π; x ∈ R). 2

Also, apply this inequality to show that Z∞ 

sin x x

2 dx =

π . 2

0

(Neville [858, pp. 629–630]) 22. Prove that Z∞ 0

  ex log x π 1 1−2a 0(1 − a) dx = log (2π) 2 sin (2π a) 0(a) e2x − 2 ex cos (2π a) + 1

(0 < a < 1).

(Zhang and Williams [1252, p. 377])

Introduction and Preliminaries

117

23. Prove that π

Z2 π 4

    0 43 √ π log (log (tan x)) dx = log    2π  . 2 0 1 4

(cf. Vardi [1189, p. 308]; see also Gradshteyn and Ryzhik [505, p. 532]) 24. Let ∞ Y (n − a1 ) (n − a2 ) · · · (n − ak ) P= (n − b1 ) (n − b2 ) · · · (n − bk )



k X



n=1

aj =

j=1

k X

 bj  ,

j=1

where no aj or bj is a positive integer. Show that

P=

k Y 0 1 − bj j=1

0 1 − aj



.

(cf. Rainville [959, pp. 13–15]; see also Melzak [821, p. 101]) 25. Let Mn and mn denote the maximum and minimum of the Bernoulli polynomial Bn (x) for 0 5 x 5 1. Show that M4n = B4n

  1 2

  = 1 − 21−4n |B4n |

M4n+2 = B4n+2 (0) = B4n+2

(n ∈ N);

(n ∈ N0 ) ;

m4n = B4n (0) = − |B4n | (n ∈ N);     m4n+2 = B4n+2 12 = − 1 − 2−1−4n B4n+2

(n ∈ N0 ) . (Lehmer [740, pp. 534])

26. Prove the following integrals: Z1

π4 log u [log(1 + u)]2 du = A − u 288

0

and Zπ

h  u i2 31 π 4 u log 2 cos du = A + , 2 480

0

where ! ∞ n X (−1)n−1 X 1 A= . (n + 1)2 k2 n=1

k=1

(Rutledge and Douglass [991, p. 30])

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Zeta and q-Zeta Functions and Associated Series and Integrals

27. Suppose that (a) f (z) is a meromorphic function of a complex variable z in C; (b) f (z) is analytic and vanishes nowhere on C \ Z− 0; (c) f (z) satisfies the functional equation z− 21

n

n−1 Y k=0

 f

z+k n



= (2π )

n−1 2

f (z)


z ∈ C \ Z− 0 , 1

z− where n ish an arbitrarily fixed  i positive integer greater than 1 and n 2 denotes the principal value exp z − 12 Log n . Then, show that the function f must be of the form:

    1 2mπ i f (z) = exp a z − + 0(z), 2 n−1 where a is an arbitrary complex constant and m is an arbitrary integer. (cf. Haruki [540, p. 174]; see also Theorem 1.1 (Bohr-Mollerup)) 28. Suppose that a function f : R+ → R satisfies the functional equation: f (x + 1) = x f (x)

x ∈ R+




and, moreover, that  e x r x f (x) = 1. x→∞ x 2π lim

Show that f (x) = 0(x) for all x ∈ R+ . (Kuczma [705, p. 129]) 29. Suppose that a function f : R+ → R satisfies the following properties: (a) f (x
+ 1) = x f (x); x (b) xe f (x) is decreasing for x ∈ R+ ; (c) f (1) = 1. Show that f (x) = 0(x) for x ∈ R+ . (Anastassiadis [37, p. 117]) 30. Suppose that a function f : R+ → R+ satisfies the following properties: x f (x) (y > 0); (a) f (x + 1) = x+y (b) f (x) is decreasing for x ∈ R+ ; (c) f (1) = 1y . Show that f (x) = B(x, y) for x ∈ R+ . (Anastassiadis [38, pp. 25–26]) 31. Prove that, for p ≥ 5 and p ∈ N,   1 (p) Bp ≡ − p2 (p − 1)! mod p5 , 2 (p)

where Bp denotes the generalized Bernoulli numbers defined by 1.6(22). (Carlitz [216, p. 112])

Introduction and Preliminaries

119

32. Prove that, for m ∈ N and i =

√

−1,

2 F1 (1, 1 ; 2m ; i)



m−1

ij log 2 i π X − − (−1) j j = (2m − 1) im 2m−1  2 4 2 j=1



  1 1+i  + 2j 2j − 1

and 2 F1 (1, 1 ; 2m + 1 ; i)



m−1

log 2 − 1 2−π X ij = m(1 + i) i 2  (−1) j j −i − 2 4 2 m m



j=1

  1 1−i  + . 2j 4j + 2

(Butzer and Hauss [198, p. 355]) 33. Prove that ∞ X

1


2 2n n=0 (2n + 1) n

=

 √  8 π G − log 2 + 3 . 3 3 (Borwein and Borwein [148, p. 386])

34. Prove that Z1

(2 − 2x)α (2 + 2x)β 1 − x2 

−1

− 1 2

    0 α + 12 0 β + 21 π 2α+2β . dx = h  i2 2 1 0(α + β + 1) 0 2 (Askey [74, p. 357])

35. Prove that log 0(z + 1) =

      1 1 1 1 1 1 z+ log z + + √ + log z + − √ 2 2 2 2 3 2 2 3 ∞ Z 1 1 − z − + log(2π ) + e−zt φ(t) dt, 2 2 0

where, for convenience,      1 1 1 1 1 1 t φ(t) := − + t − 1 − e− 2 t cosh √ t 2 t e −1 2t 2 3    1 t −2t √   cosh d 1−e 2 3 − ;  dt  t moreover, 1

φ(t) =

e− 2 t t



      1 1 1 t 1 t cosech t + √ sinh . √ − cosh √ 2 2 t 2 3 2 3 2 3 (cf. Watson [1212, p. 5]; see also Eq. (29))

120

Zeta and q-Zeta Functions and Associated Series and Integrals

36. Prove that, for fixed p ∈ R+ , ∞ P

γ = lim

k=0

n→∞

nk k!

!p (Hk − log n)

∞ P k=0

nk k!

,

!p

where γ is the Euler-Mascheroni constant and Hk denotes the harmonic numbers defined by 3.2(36). (Brent and McMillan [174, pp. 310]) 37. Let u be the row vector {uk = 1/k : k ∈ N} and let M be the matrix with entries {mij = 1/(i − j + 1) if j 5 i, mij = 0 if j > i : i, j ∈
N}. Let v be the column vector {vn = 1/(n + 1) : n ∈ N}. Show that the product u M−1 v exists (as a convergent series), and is equal to Euler-Mascheroni constant γ defined by 1.1(3). (Kenter [644, p. 452]) 38. Let (z) be the function of the complex variable z defined by (z) :=

0(z + a1 ) 0(z + a2 ) 0(z + b1 ) 0(z + b2 )

in which a1 , a2 , b1 , b2 are to be regarded as any constants (real or complex). Show that, for | arg z| < π and large |z|, (z) ∼

  p  c2 c3 c1 1 + + + ··· , 1+ z z + 1 (z + 1)(z + 2) (z + 1)(z + 2)(z + 3)

where the cn are constants and p = b1 + b2 − a1 − a2 . (Engen [1183, pp. 124–125]) 39. Prove the following Lipschitz summation formula: ∞ X

e−π i λ/2 π λ X λ−1 π i τ n n e 0(λ) ∞

(2m + τ )−λ =

m=−∞

(<(τ ) > 0; λ > 1).

n=1

(Knopp [677, p. 65]) 40. Prove the following series-integral representation of the Catalan constant G:  π 2 Z2 ∞ 1X   G= (−1)n  cos n t dt . 2 n=0

0

(Catalan [224, p. 51]) 41. Prove that
2 1 0 ψ (x) ψ 000 (x) < ψ 00 (x) 2

(x > 0),

where ψ(x) denotes the Psi (or Digamma) function. (cf. English and Rousseau [419, p. 432]; see also Alzer and Wells [33, p. 1459])

Introduction and Preliminaries

121

42. A function f is said to be completely monotonic on an interval I, if f ∈ C∞ (I) and (−1)k f (k) (x) = 0

(∗)

for all x ∈ I and for all k ∈ N0 . If the inequality (∗) is strict for all x ∈ I and for all k ∈ N0 , then f is said to be strictly completely monotonic on I. Consider the function  2 Fn (x; c) := ψ (n) (x) − c ψ (n−1) (x) ψ (n+1) (x)
x ∈ R+ ; c ∈ R; n ∈ N \ {1} . Let n ∈ N \ {1}, and let α, β ∈ R. Show that the function. x 7→ Fn (x; α) is strictly completely monotonic on (0, ∞), if and only if α5

n−1 , n

and that the function: x 7→ −Fn (x; β) is strictly completely monotonic on (0, ∞), if and only if β=

n . n+1 (Alzer and Wells [33, p. 1460])

43. Let the function g(x) be defined by g(x) := x2 ψ 0 (1 + x) − x ψ(1 + x) + log 0(x + 1)

(x > −1).

Show that g(x) strictly decreases from ∞ to 0 on (−1, 0] and strictly increases from 0 to ∞ on [0, ∞). (Elbert and Laforgia [404, Theorem 2]) 44. Prove that 1

0(x) xx− 2 ey xx−1 ey 5 5 y−1 x 0(y) yy− 12 ex y e


x=y>1 . (Ke˘cki´c and Vasi´c [642, p. 107])

45. Prove that (a) For n, an odd positive integer, G = −T

    n 1 2n + 1 X 2j − 1 = (−1) j T ; 4 n+1 8n + 4 j=1

(b) For n, an even positive integer, G = −T

    n 1 2n + 1 X 2j − 1 = (−1) j+1 T , 4 n 8n + 4 j=1

122

Zeta and q-Zeta Functions and Associated Series and Integrals

where, for convenience, T is defined by Zr π T(r) :=

log(tan θ ) dθ



 1 . 05r5 2

0

(Bradley [165, pp. 164–165]) 46. For the Stirling numbers s(n, k) defined by 1.6(1), show that ∞ X (−1)n (1 − a)k (k) (a) s(n + k, k) = − ψ (a − k); (n + k)(a)n k! n=0

∞ X

(−1)n (−a)k s(n + k, k) = ψ (k−1) (a − k). (n + k − 1)(a)n a(k − 1)! n=0 Also deduce the special cases of each of these summation formulas, when a = k + 1 and a = k + 2.

(b)

(Cf. Jordan [614, p. 343]; see also Hansen [531, p. 348]) 47. Verify that the special case of Problem 46(a) above, when a = k + 1 is precisely the relationship 3.5(16). (Cf. Problem 5 (Chapter 3)) 48. Show that Z∞ 0

π cos x dx = xp 2 0(p) cos (p π/2)

(0 < p < 1)

and Z∞ 0

sin x π dx = xp 2 0(p) sin (p π/2)

(0 < p < 1). (Andrews [49, p. 83 and p. 85])

49. For p ∈ N, prove the following integral: Z1

Z1 ···

0



p Y

 j=1

0

=

p Y ν=1

y−1 x−1  p Y  |1(u)|2z du1 · · · dup (1 − uj ) uj    j=1

0(1 + νz)0(x + (ν − 1)z) 0(y + (ν − 1)z) 0(1 + z) 0(x + y + (p + ν − 2)z)

   1 <(x) <(y) <(x) > 0; <(y) > 0; <(z) > − min , , , p p−1 p−1 where 1(u) = 1(u1 , u2 , . . . , up ) :=

Y


uj − ui .

i<j

(Selberg [1013]; see aso [1015, pp. 204–213])

Introduction and Preliminaries

123

50. Prove the result asserted by Equation 1.3(57) in two ways, as noted there. (Choi and Cvijovi´c [271, Theorem 1]) 51. Show that the generalized Barnes G-function Gm defined by  m  X ζ ( j − m) γ + Hm Gm (1 + z) = exp − (−z) j − (−z)1+m  j m+1 j=1  m    X m 0 + ζ (j − m) (−z) j  j j=1

∞  Y

(

·

1+

k=1

m+1 X

z k m exp k

r=1

! !) 1 r (−z) km r kr

(m ∈ N0 )

is an entire function of order m with the zeros-k with multiplicity km (k ∈ N). Moreover, there is the zero 0 with multiplicity 1 for m = 0. The used infinite product is absolutely convergent. One has Gm (1) = 1, and Gm satisfies the functional equation Gm (z + 1) =

m Y

r m {Gm−r (z)}(−1) ( r )

(m ∈ N).

r=0

Here, as usual, ζ (s) is the Riemann zeta function, γ the Euler-Mascheroni constant, and Hm the harmonic numbers. (Schuster [1010, Theorem A]) 52. Show that the following integrals: Ik (q) :=

Z∞

t
k+1 1 + t2

0

(k ∈ N)


dt

e2πqt − 1

are given by   k 2k
1 1 X (−1) j+1 2k − j − 1 j−1 j (j) k Ik (q) = − − 2k+2 + 2k 2 q ψ (q). 4k 2 ( j − 1)! k−j q k2 j=1

(See Boros et al. [141, Theorem 3.2]) 53. For arbitrary n ∈ N, show that 1 2n +

1 1−γ

−2

5 Hn − log n − γ <

1 2n +

1 3

,

where γ is the Euler-Mascheroni constant and Hn the harmonic numbers. Here, the con1 stants 1−γ − 2 and 13 are the best possible. (See Qi et al. [938, Theorem 2]) 54. The Genocchi numbers Gn are defined by means of the following generating function: ∞

∞

n=0

n=1

X X 2t tn t2n =: Gn = t + G2n , t e +1 n! (2n)!

124

Zeta and q-Zeta Functions and Associated Series and Integrals

which are directly related to the Bernoulli numbers Bn by means of   Gn = 2 1 − 22n Bn . Show that G2n = −n −

n−1   1 X 2n G2k 2k 2

and

G2n = −1 −

k=1

 n−1  X 2n G2k . 2k − 1 2k k=1

(See Herrmann [555, Proposition 2.1]) 55. Let α > 0 and 0 < c 6= 1 be real numbers. Show that the following inequality:       0(x + y + c) 1/α 0(x + c) 1/α 0(y + c) 1/α < + 0(x + y) 0(x) 0(y) holds true for all positive real numbers x and y, if and only if α = max {1, c}. The reverse inequality is valid for all x, y > 0, if and only if α 5 min {1, c}. (See Alzer [27]) 56. Hadamard’s Gamma function H(x) is defined by    d 0(1/2 − x/2) 1 log , H(x) = 0(1 − x) dx 0(1 − x/2) where 0(x) denotes the classical Euler’s Gamma function given in 1.1(1). H(x) is an entire function and satisfies the following relationship: H(n) = (n − 1)!

(n ∈ N).

Show that H can be expressed as follows:       sin (πx) x x+1 H(x) = 0(x) 1 + ψ −ψ 2π 2 2 in terms of the sine, Gamma and Psi (or Digamma) functions. Furthermore, Hadamard’s Gamma function H(x) satisfies the following remarkable functional equation, which is a counterpart of 1.1(9): H(x + 1) = x H(x) +

1 , 0(1 − x)

which is a counterpart of 1.1(9). (See Alzer [28]; Luschny [792]; Newton [859]) 57. Show that the following inequalities hold true:  h i2  m+n ψm (x) ψn (x) = ψ m+n (x) x > 0; m, n, ∈N 2 2 and (s + 1)

ζ (s) ζ (s + 1) =s ζ (s + 1) ζ (s + 2)

(s > 1) ,

where ψn (x) := ψ (n) (x) and ζ (s) is the Riemann Zeta function given in 2.3(1). (See Laforgia and Natalini [725])

Introduction and Preliminaries

125

58. Show that

γ=

  r ∞ 1 2n X 2r n X 1 − n ln 2 + O , n n (r + 1)! s+1 e2 2n e2 r=0

s=0

where γ denotes the Euler-Mascheroni constant (see Section 1.2). (See Bailey [84]; Mortici [846]) 59. Let m ∈ N and z ∈ C. Suppose also that 1(m; λ) abbreviates the following array of m parameters: λ+m−1 λ λ+1 , ,..., m m m

(m ∈ N).

Show that n X

  n  (2mk + 1)!! zk 1 X 2mk + 1 n  z k = (n − k)!(mk + 1)! k! n! mk k 2m k=0 k=0     −n, 1 m; 23 ; 1   = − 2 m · z . m+1 Fm  n! 1 (m; 2) ;

(37)

(Sofo and Srivastava [1044]; see, for special cases, Srivastava [1089] and Samoletov [1000]) 60. Show that each of the following limit formulas holds true:

lim

0(nz) (−1)(n−q)k · q (qk)! = · 0(qz) n (nk)!

lim

ψ(nz) q = ψ(qz) n

z→−k

(n, q ∈ N; k ∈ N0 )

and

z→−k

(n, q ∈ N; k ∈ N0 ).

(Prabhu and Srivastava [912]) 61. For a suitably bounded sequence {Cn }n∈N0 of essentially arbitrary complex numbers, show that ∞ X n=0

[n/N]   ∞ Cn ωn zα+Nn X γ (α + n, z) X n (Nk) ! = Ck ωk α + Nn n! n! Nk k! n=0

k=0

(N ∈ N) ,

126

Zeta and q-Zeta Functions and Associated Series and Integrals

provided that each member exists. Also, deduce the following hypergeometric form of this expansion formula:
 α/N, αp ;  N z  z p+1 Fq+1 
ω −N (α + N) /N, βq ; 

α

=α

∞ X γ (α + n, z)

n!

n=0


 1 (N; −n) , αp ;  p+N Fq 
ω βq ; 

(N ∈ N) .

(Lin et al. [764, p. 518]; see, for special cases, Gautschi et al. [477]) 62. For a suitably bounded sequence { (n)}n∈N0 of essentially arbitrary complex numbers, show that ∞ X

n X

 (k)

k=0 k1 ,··· ,kr =0

(−n)k1 . . . (−n)kr (γ + n)k+k1 +···+kr k1 ! · · · kr ! (γ )k+k1 +···+kr ·

=

(β1 )k1 . . . (βr )kr (β1 + 1)k1 · · · (βr + 1)kr

∞ (γ − β1 − · · · − βr + n)k (n!)r (γ − β1 − · · · − βr )n X ,  (k) (γ )n (β1 + 1)n . . . (βr + 1)n (γ − β1 − · · · − βr )k k=0

provided that the series involved are absolutely convergent. (Carlitz [219, p. 169]; see also Lin and Srivastava [766, p. 310] and Problem 26 of Chapter 6) (s) 63. Let the harmonic numbers Hn and the generalized harmonic numbers Hn be defined, as usual, by Hn := Hn(1)

and

Hn(s) :=

n X 1 ks

(s ∈ C; n ∈ N := {1, 2, 3, . . .}),

k=1

(a) For αj , βj = 0 (j = 1, 2, 3, 4) and m ∈ N0 , show that   n+m−1 [ψ(β1 + 1 + α1 n) − ψ(β1 + 1)] ∞ X n−1   4 Q α i n + βi n=1 n4 βi i=1 tn

= −α1 α2 α3 α4

Z1 Z1 Z1 Z1

(1 − x)β1 ln (1 − x) · (1 − y)β2 (1 − z)β3 (1 − w)β4 xyzw

0 0 0 0

·

x α1 y α2 z α3 w α4 (1 − tx α1 y α2 z α3 w α4 )m+1

dx dy dz dw.

Introduction and Preliminaries

127

(b) Let p and q be positive integers. In terms of the Bernoulli polynomials Bn (x) and the generalized Clausen functions Cln (z) , show that        2πps   n   cos         q q  (−1) 2 (2π )n+1 P s   Bn+1        2 (n + 1) ! q    s=1  2π ps p    (n)     ψ  sin      q  q   n          = n!q     2π ps p    (n)      ψ      sin      q q q   P 2πs   Cln+1 ±      q   s=1   2π ps     cos  q 

  n = 2m − 1 ; m ∈ N; 1 5 p 5 q , n = 2m

where the Bernoulli polynomials Bn (x) are generated, as usual, by ∞

X tn text = B (x) n et − 1 n!

(|t| < 2π )

n=0

and the generalized Clausen functions Cln (θ ) are given by

Cln (θ ) =

 ∞ P sin (kθ )      k=1 kn

(n

even)

 ∞ cos (kθ )  P    kn k=1

(n

odd).

(c) Let α1 = 0, α2 = 0, α3 = 0

and

α4 = 0

be positive real numbers. Also, let p ∈ N0 and j, k, l, m ∈ N0 , 0 5 p 5 3k − 4

and

|t| 5 1.

Then show that ∞ X

np tn [ψ( j + 1 + α1 n) − ψ( j + 1)]      α1 n + j α2 n + k α3 n + l α4 n + m n=1 j k l m Z1 Z1 Z1 Z1

(1 − x) j (1 − y)k−1 (1 − z)l−1 (1 − w)m−1 x 0 0 0 0
· ln (1 − x) Li−p−1 txα1 yα2 zα3 wα4 dx dy dz dw,

= −α1 klm

128

Zeta and q-Zeta Functions and Associated Series and Integrals

where the Polylogarithmic function Liq (β) is given by Liq (z) =

∞ r X z r=1

rq

.

(See Sofo and Srivastava [1045]) 64. Let [τ ] denote the greatest integer in τ ∈ R. Then, for an essentially arbitrary sequence {n }n∈N0 of complex numbers, show that the following general combinatorial series relationship holds true: [n/m] X k=0

 [k/m]   [n/m] X λ + n + 1λ + mk λ+k X k  k zk  j zj = n − mk mk k mj j=0

k=0

(λ ∈ C; m ∈ N; n ∈ N0 ) or, equivalently, [n/m] X k=0

   [k/m]     [n/m] n λ+k X k λ+n+1 X λ+1  k zk  j zj = λ + mk + 1 mk k mj n j=0

k=0

(λ ∈ C; m ∈ N; n ∈ N0 ), provided that both members of each of these assertions exist. Also, deduce the following general Fox-Wright hypergeometric series relationship:   (−n, m), (α, m), (a1 , A1 ), . . . , (ap , Ap ); ∗ z p+3 9q+1  (α + 1, m), (b1 , B1 ), . . . , (bq , Bq );   −1 [n/m] (−k, m), (a1 , A1 ), . . . , (ap , Ap ); X α + k − 1 ∗ z , p+1 9q k (b1 , B1 ), . . . , (bq , Bq ); k=0

α+n = n 

provided that each member of this assertion exists. Here, the Fox-Wright function ∗ p 9q (p, q ∈ N0 ) or p 9q (p, q ∈ N0 ), with p numerator parameters a1 , · · · , ap and q denominator parameters b1 , · · · , bq , such that aj ∈ C ( j = 1, . . . , p)

and

bj ∈ C \ Z− 0 ( j = 1, . . . , q),

is defined by
 
∞ (a ) (a1 , A1 ) , . . . , ap , Ap ; X 1 A1 k . . . ap Ap k zk ∗ 
p 9q
z := (b ) . . . bq Bq k k! (b1 , B1 ) , . . . , bq , Bq ; k=0 1 B1 k
 
(a1 , A1 ) , . . . , ap , Ap ; 0(b1 ) · · · 0 bq 
p 9q  =
z 0(a1 ) · · · 0 ap (b1 , B1 ) , . . . , bq , Bq ;   q p X X Aj > 0 ( j = 1, . . . , p) ; Bj > 0 ( j = 1, . . . , q) ; 1 + Bj − Aj = 0 , j=1

j=1

Introduction and Preliminaries

129

where the equality in the convergence condition holds true for suitably bounded values of |z|, given by     p q Y Y −Aj Bj |z| <  Aj  ·  Bj  . j=1

j=1

Clearly, in terms of the generalized hypergeometric function p Fq (p, q ∈ N0 ), we have the following relationship (see, for details, Section 1.5):
    a1 , . . . , ap ; (a1 , 1) , . . . , ap , 1 ; ∗   z p 9q
z = p Fq b1 , . . . , bq ; (b1 , 1) , . . . , bq , 1 ;
 (a1 , 1) , . . . , ap , 1 ; 
p 9q  =
z . 0(a1 ) · · · 0 ap (b1 , 1) , . . . , bq , 1 ; 0(b1 ) · · · 0 bq






(See R. Srivastava [1118]) 65. For the Bernoulli numbers Bn , which are usually given by the recurrence relation 1.7(6), that is, by Bn = (−1)n

n   X n Bk k

(n ∈ N0 )

k=0

and Bn = −

 n−1  1 X n+1 Bk n+1 k

(n ∈ N),

k=0

derive each of the following computationally more advantageous recursion formulas: B2n = −

 n−1  X n+1 1 (n + k + 1)Bn+k (n + 1)(2n + 1) k

(n ∈ N)

k=0

or, equivalently,  n−1  1 X n+1 Bn = − Bn+k n+1 k


Bn := (n + 1)Bn ; n ∈ N ,

k=0

B2n =

 n−1  X 1 2n + 2 1 − B2k 2n + 1 (n + 1)(2n + 1) 2k

(n ∈ N)

k=0

and B2n =

 n−1  X 1 1 2n + 1 − B2k 2 2n + 1 2k

(n ∈ N).

k=0

(Cf. Srivastava and Miller [1104]; see also [625])

130

Zeta and q-Zeta Functions and Associated Series and Integrals

n o ( j) 66. In terms of the sequences an

fj (z) =

∞ X

( j)

an zn

n∈N0

( j = 1, . . . , r), let

( j = 1, . . . , r).

n=0

Denote also the familiar multinomial coefficient by 

 n n! := n1 ! . . . nr ! n1 , . . . , nr

(n, nj ∈ N0 ; j = 1, . . . , r; r ∈ N).

By applying the following known series relationship involving product of power series: r Y

{ fj (z)} =

j=1

∞ X

bn zn

n=0

 X

bn :=

( ( j) ) r Y anj , nj !

n1 +···+nr =n j=1

(α)

or otherwise, derive several properties of the generalized Bernoulli polynomials Bn (x) (α) and the generalized Euler polynomials En (x) of order α as follows: 1 +···+αr ) B(α (x1 + · · · + xr ) = n

n1 +···+nr =n

En(α1 +···+αr ) (x1 + · · · + xr ) =

Y r n
o n (α ) Bnj j xj , n1 , . . . , nr



X

j=1



X n1 +···+nr =n

Y r n
o n (α ) Enj j xj n1 , . . . , nr j=1

and n   X n k=0

k

(α) (α) Bn−k (x)Ek (y) = 2n

B(α) n



 x+y . 2

(Cf. Brychkov [193]) 67. In terms of the sequence {n }n∈N0 , let the function 8(z) be defined by 8(z) =

∞ X

 k zk

(|z| < R; R ∈ R+ ).

k=0

Suppose also that ω = exp



2πi n

 and

m ∈ {0, 1, . . . , n − 1} (n ∈ N).

Introduction and Preliminaries

131

Then show that ∞ X

nk+m znk+m =

k=0

n−1

n

k=0

k=1

1 X (n−m)k  k  1 X (n−m)k  k  ω 8 ω z = ω 8 ω z n n

(|z| < R)

and apply these identities to derive the corresponding results involving the Fox-Wright function p 9q (p, q ∈ N0 ) or p 9q∗ (p, q ∈ N0 ) (see Problem 64 above). (See, for details, Srivastava [1081]) 68. In connection with the so-called Littlewood’s teaser about the hitherto nonexistent formula for the sum: n  3 X n r=0

r

,

show that n  k X n r=0

r

2kn ∼√ k



2 πn

 1 (k−1) 2

(n → ∞; k ∈ N)

and n  k X n r=0

r

x n−r yr =

[n/2] X r=0

n 2r

   2r n + r r r x y (x + y)n−2r . r r

(See P´olya and Szeg¨o [906, p. 65, Theorem 40; p. 239, Entry 40] and MacMahon [794, p. 122]; see also Nanjundiah [854] and Srivastava [1075]) 69. For the Bernoulli polynomials Bn (x), show that Bn (x) =

n X k=0

  k   1 (µ + 1)k − x X k (−1) j (µk + j)n k+1 k j j=0

+ (µ + x)

n X k=0

    k X 1 k (µ + 1)k − x (−1) j Gk (µ; x) (µk + j)n , k+1 k j j=0

where, for convenience, G0 (µ; x) = 0

and

Gk (µ; x) =

k X `=1

1 x − µk − `

(k = 1, . . . , n; µ ∈ C).

(See Srivastava [1074, p. 81, Eq. (27)]) 70. Show that the following relationship holds true: (l)

Gn(l) (x; λ) = {n}l En−l (x; λ) =

n! (l) E (x; λ) (n − l)! n−l


n, l ∈ N0 ; 0 5 l 5 n; λ ∈ C

132

Zeta and q-Zeta Functions and Associated Series and Integrals

or, equivalently,

En(l) (x; λ) =

n! 1 (l) (l) Gn+l (x; λ) = G (x; λ) {n + l}l (n + l)! n+l

(n, l ∈ N0 ; λ ∈ C)

between the Apostol-Genocchi polynomial of order l and the Apostol-Euler polynomial of order n − l, which are defined, respectively, by 1.8(58) and 1.8(15) with α = l (l ∈ N0 ). (See Luo and Srivastava [791, Lemma 2]) 71. Derive the following relationship: Gn(α) (x; λ) = (−2)α Bn(α) (x; −λ)

α, λ ∈ C; 1α := 1




or, equivalently,

Bn(α) (x; λ) =

1 G (α) (x; −λ) (−2)α n

α ∈ C; 1α := 1




(α)

between the Apostol-Genocchi polynomials Gn (x; λ) and the Apostol-Bernoulli polyno(α) mials Bn (x; λ), which are defined by 1.8(58) and 1.8(13), respectively. (See Luo and Srivastava [791, Lemma 3]) 72. Verify that the following relationship holds true:

Bn(l) (x; λ) =

n! (l) E (x; −λ) (n − l)!(−2)l n−l


n, l ∈ N0 ; 0 5 l 5 n; λ ∈ C

or, equivalently,

En(l) (x; λ) =

n!(−2)l (l) B (x; −λ) (n + l)! n+l

(n, l ∈ N0 ; λ ∈ C)

between the Apostol-Bernoulli polynomial of order l and Apostol-Euler polynomial of order l, which are defined, respectively, by 1.8(13) and 1.8(15), with α = l (l ∈ N0 ). (See Luo and Srivastava [791, Lemma 4]) (α) 73. For the generalized Bernoulli polynomials Bn (x; λ; a, b, c) of order α ∈ C, defined by 1.8(66), derive each of the following identities:

B(α) n (x + 1; λ; a, b, c) =

n   X n

(α)

(ln c)n−k Bk (x; λ; a, b, c) ; k k=0   a b (α) B(α) + α; λ; a, b, c) = B x; λ; , , c ; (x n n c c

Introduction and Preliminaries

133

  a b (α) B(α) − x; λ; a, b, c) = B −x; λ; , , c ; (α n n c c

αλ ln

 X n   n b (α+1) (ln b)k Bn−k (x; λ; a, b, c) = (α − n) B(α) n (x; λ; a, b, c) k a k=0

(α)

+ n (x ln c − α ln a) Bn−1 (x; λ; a, b, c) or, equivalently,

α ln

 X n   b n (α+1) (ln a)k Bn−k (x; λ; a, b, c) = (α − n) B(α) n (x; λ; a, b, c) a k k=0

(α)

+ n (x ln c − α ln b) Bn−1 (x; λ; a, b, c) ; k   X k (α+β) (β) (α) Bk Bk−r (x; λ; a, b, c) Br (y; λ; a, b, c) ; (x + y; λ; a, b, c) = r r=0

(α)

Bk (x + y; λ; a, b, c) =

k  X r=0

 k (α) (y ln c)r Bk−r (x; λ; a, b, c) ; r

o n! ∂ l n (α) (α) B λ; a, b, c) = (x; (ln c)l Bn−l (x; λ; a, b, c) n (n − l) ! ∂xl

(l ∈ N0 )

and Zη

B(α) n (x; λ; a, b, c) dx

ξ

=

h i 1 (α) (α) Bn+1 (η; λ; a, b, c) − Bn+1 (ξ ; λ; a, b, c) (n + 1) ln c

(η > ξ ).

(See, for details, Srivastava et al. [1095]) (α) 74. For the generalized Bernoulli polynomials Bn (x; λ; a, b, c) of order α ∈ C, defined by 1.8(66), show that each of the following explicit series representations holds true:

 X   n−l  n n−l l+r−1 λr (x ln c − l ln a)n−r−l r+l l r r − 1) (λ r=0         r r j ln ba X r b  · j ln (−1) j 2 F1 l + r − n, 1; r + 1; − j a x ln c − l ln a

Bn(l) (x; λ; a, b, c) = l!

j=0

a, b, c ∈ R+ (a 6= b); l ∈ N0 ; λ ∈ C \ {1}




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Zeta and q-Zeta Functions and Associated Series and Integrals

and ln c ln λ

−x ln b−ln a B(l) (ln b − ln a)n−l n (x; λ; a, b, c) = e  r   ∞ X n+k X ln a k · (−l)r ln b − ln a r


k=0 r=0

     n+k−r  n + k − r − l n + k − r −1 (ln λ)k X n + k − r l + p − 1 p! · k k−r k! p p (2p) ! p=0

p X

   n+k−r−p p 2p ln c j x · +j (−1)j j ln b − ln a j=0     j ln ba   · 2 F1 p − n − k + r, p − l; 2p + 1; x ln c + j ln ba
a, b, c ∈ R+ (a 6= b); l ∈ N0 , where 2 F1 (a, b; c; z) denotes the Gauss hypergeometric function, defined by 1.5(4). (Srivastava et al. [1095, p. 258, Theorem 6; p. 260, Theorem 7]) 75. For the generalized Apostol type polynomials Yn,β (x; k, a, b), defined by 1.8(61), derive each of the following properties: d  Yn,β (x; k, a, b) = nYn−1,β (x; k, a, b), dx

anb(m−1) mv−k

m−1 X j=0

β a

= amb(n−1) nv−k

bjn

n−1  blm X β l=0

Zy 0

 Yv,β m

a

x nj + ; k, am , b m m 

Yv,β n



 x ml + ; k, an , b , n n

Y n+1,β (y; k, a, b) − Yn+1,β (k, a, b)   n+1 Yn,β (x; k, a, b)dx =   0

(n ∈ N) (n = 0)

and !b   N 1 k−1 n−1 X β j−1 Yn+k−1,β (x; k, a, b) = − N 2 a j−N j=1   x+j−1 · Yn+k−1,β N ; k, aN , b . N (See, for details, Ozden et al. [886])

Introduction and Preliminaries

135 (α)

76. For the generalized Euler polynomials En (x; λ; a, b, c) of order α ∈ C, defined by 1.8(68), derive each of the following identities:

E(α) n (x + 1; λ; a, b, c) =

n   X n

(α)

(ln c)n−k Ek (x; λ; a, b, c) , k k=0   a b (α) (α) En (x + α; λ; a, b, c) = En x; λ; , , c , c c   a b (α) , , c , E(α) − x; λ; a, b, c) = E −x; λ; (α n n c c  c c  n (α) E(α) x; λ; , , c , n (α − x; λ; a, b, c) = (−1) En a b  X n   α b n (α+1) (α) ln (ln a)k En−k (x; λ; a, b, c) = En+1 (x; λ; a, b, c) 2 a k k=0

− (x ln c − α ln b) E(α) n (x; λ; a, b, c) ,  X k n   b n b αλ (α+1) (α) ln En−k (x + 1; λ; a, b, c) = En+1 (x; λ; a, b, c) ln 2 a c k k=0

+ (x ln c − α ln a) E(α) n (x; λ; a, b, c) ,  X n   αλ b n (α+1) (α) ln (ln b)k En−k (x; λ; a, b, c) = En+1 (x; λ; a, b, c) 2 a k k=0

− (x ln c − α ln a) E(α) n (x; λ; a, b, c) ,   n X n (α) (α+β) (β) En E (x + y; λ; a, b, c) = (x; λ; a, b, c) Ek (y; λ; a, b, c) , k n−k k=0 n   X n (α) (α) En (x + y; λ; a, b, c) = (y ln c)n−k Ek (x; λ; a, b, c) , k k=0

∂l ∂xl

n

o

E(α) n (x; λ; a, b, c) =

n! (α) (ln c)l En−l (x; λ; a, b, c) (n − l) !

and Zη ξ

E(α) n (x; λ; a, b, c) dx =

h i 1 (α) (α) En+1 (η; λ; a, b, c) − En+1 (ξ ; λ; a, b, c) (n + 1) ln c

(η > ξ ),

it being understood (wherever needed) that a, b, c ∈ R+ (a 6= b), x ∈ R

and

l ∈ N0 . (See, for details, Srivastava et al. [1096])

136

Zeta and q-Zeta Functions and Associated Series and Integrals (α)

77. For the generalized Euler polynomials En (x; λ; a, b, c) of order α ∈ C, defined by 1.8(67), show that each of the following explicit series representations holds true: α E(α) n (x; λ; a, b, c) = 2

 n   X n α+k−1 k=0

·

k X j=0

k

k

λk (λ + 1)α+k   j ln ba

  k b ln (x ln c − α ln a)n−k a 

   k k  j 2 F1 k − n, 1; k + 1; − (−1) j x ln c − α ln a j


a, b, c ∈ R+ (a 6= b); α ∈ C; λ ∈ C \ {−1} ,    k X   n   k X n α+k−1 λk b α j k k E(α) ln j (−1) n (x; λ; a, b, c) = 2 k k j a (λ + 1)α+k j=0 k=0     b   n−k j ln a b   · x ln c − α ln a + j ln 2 F1 k − n, k; k + 1; − a x ln c − α ln a + j ln ba
a, b, c ∈ R+ (a 6= b); α ∈ C; λ ∈ C \ {−1} and −x E(α) n (x; λ; a, b, c) = e

ln c ln λ ln b−ln a




ln

 k X   j ∞ X n b n ln a (−α) j j a ln b − ln a k=0 j=0

k n+k−j X 

   m   m m ` n+k−j α+m−1 X (ln λ) · (−1)` ` m m k! 2 m=0 `=0      n+k−j−m ` ln ba x ln c  , · +` 2 F1 m − n − k + j, m; m + 1; ln b − ln a x ln c + ` ln ba
a, b, c ∈ R+ (a 6= b); α ∈ C , where 2 F1 (a, b; c; z) denotes the Gauss hypergeometric function, defined by 1.5(4). (Srivastava et al. [1096, pp. 294–295, Theorem 6; p. 298, Theorem 7]) 78. Show that the following relationships hold true: α (α) G(α) n (x; λ; a, b, c) = (−2) Bn (x; −λ; a, b, c)

α ∈ C; 1α := 1




and (l)

n! (l) E (x; λ; a, b, c) (n − l)! n−l
n, l ∈ N0 ; n = l; λ ∈ C

l G(l) n (x; λ; a, b, c) = (−1) (−n)l En−l (x; λ; a, b, c) =

(α)

between the generalized Bernoulli polynomials Bn (x; λ; a, b, c), the generalized (α) Euler polynomials En (x; λ; a, b, c) and the generalized Genocchi polynomials (α) Gn (x; λ; a, b, c) of order α ∈ C, defined by 1.8(66), 1.8(67) and 1.8(68), respectively. (Srivastava et al. [1096, p. 300, Lemma 3])

Introduction and Preliminaries

137 (α)

79. For the generalized Genocchi polynomials Gn (x; λ; a, b, c) of order α ∈ C, defined by 1.8(69), show that each of the following explicit series representations holds true:     k n−l  λk b 2l · n! X n − l k + l − 1 ln k+l k k (n − l)! a + 1) (λ k=0       k j ln ba X k  · (x ln c − l ln a)n−k−l jk 2 F1 k + l − n, 1; k + 1; − (−1) j j x ln c − l ln a

G(l) n (x; λ; a, b, c) =

j=0

a, b, c ∈ R+ (a 6= b); l ∈ N0 ; λ ∈ C \ {1}




and G(l) n (x; λ; a, b, c) =

  k   n−l  2l · n! X n − l k + l − 1 b λk ln k+l (n − l)! k k a (λ + 1) k=0

k X

    n−k−l k k b · j x ln c − l ln a + j ln (−1) j j a j=0     j ln ba   · 2 F1 k + l − n, k; k + 1; − x ln c − l ln a + j ln ba
a, b, c ∈ R+ (a 6= b); l ∈ N0 ; λ ∈ C \ {1} , where 2 F1 (a, b; c; z) denotes the Gauss hypergeometric function, defined by 1.5(4). (Srivastava et al. [1096, p. 301, Theorem 9]) 80. For the Apostol-Genocchi polynomials Gn (x; λ) (λ ∈ C), defined by 1.8(58) (with α = 1), derive the following exponential Fourier series representations: ∞ e(2k−1)πix 2 · n! X x [(2k − 1)π i − log λ]n λ k=−∞  

! ∞ ∞ X exp − nπ (2 · n!)in X exp nπ 2 − (2k + 1)π x i 2 + (2k + 1)π x i = + [(2k + 1)π i + log λ]n [(2k + 1)π i − log λ]n λx

Gn (x; λ) =

k=0

k=0

(n ∈ N; 0 5 x 5 1; λ ∈ C \ {0, −1}). (See Luo and Srivastava [791, p. 5726, Theorem 20]; see also [786]) (α) 81. Derive each of the following properties of the Apostol-Genocchi polynomials Gn (x; λ) of order α ∈ C, defined by 1.8(58): Gn(α) (λ) = Gn(α) (0; λ) ,

Gn(0) (x; λ) = x n ,

Gn(0) (λ) = δn,0

G0 (x; λ) = G0 (λ) = δα,0

and

(α)

(α)

(n ∈ N0 ; α ∈ C),

138

Zeta and q-Zeta Functions and Associated Series and Integrals

where δn,k denotes the Kronecker symbol; n   X n (α) G (λ) x n−k Gn(α) (x; λ) = k k k=0

and Gn(α) (x; λ) =

n   X n k=0

k

(α−1)

Gn−k (λ)Gk (x; λ);

λGn(α) (x + 1; λ) + Gn(α) (x; λ) = 2n o ∂ n (α) (α) Gn (x; λ) = nGn−1 (x; λ) ∂x

(α−1)

Gn−1 (x; λ)

(n ∈ N);

(n ∈ N)

and o ∂ p n (α) n! (α) Gn (x; λ) = G (x; λ) p ∂x (n − p)! n−p Zb

Gn(α) (x; λ)dx =

(α)

(n, p ∈ N0 ; 0 5 p 5 n);

(α)

Gn+1 (b; λ) − Gn+1 (a; λ) n+1

a

and Zb

Gn(α) (x; λ)dx =

n X k=0

a (α+β)

Gn

(x + y; λ) =

  1 n (α) G (λ) (bn−k+1 − an−k+1 ); n−k+1 k k

n   X n (α) (β) G (x; λ)Gn−k (y; λ); k k k=0

(−1)n+α (α) Gn(α) (α − x; λ) = Gn (x; λ−1 ) λα and (−1)n+α (α) Gn (−x; λ−1 ); λα αλ (α+1) (α) (n − α) Gn(α) (x; λ) = nx Gn−1 (x; λ) − G (x + 1; λ) 2 n Gn(α) (α + x; λ) =

and

α (α+1) (α) (x; λ) = n(α − x) Gn−1 (x; λ) + (n − α) Gn(α) (x; λ). G 2 n

(See Luo and Srivastava [791, pp. 5706–5607]) (α) 82. For the Apostol-Genocchi polynomials Gn (x; λ) of order l (l ∈ N0 ), defined by 1.8(58) with α = l (l ∈ N0 ), show that the following explicit series representations hold true:  X   n−l  n n−l l+k−1 λk Gn(l) (x; λ) = 2l l! k k l (λ + 1)l+k k=0 ·

k X j=0

(−1) j

    k k j j (x + j)n−k−l 2 F1 l + k − n, k; k + 1; j x+j (n, l ∈ N0 ; λ ∈ C \ {−1})

Introduction and Preliminaries

139

and Gn(l) (x; λ) = e−x log λ

    ∞  X n + k − l n + k −1 n + k l! (log λ)k l k k k! k=0

n+k−l X

   1 n+k−l l+r−1 r r 2r r=0     r X r r j j (x + j)n+k−r−l 2 F1 r + l − n − k, r; r + 1; · (−1) j j x+j ·

j=0

(n, l ∈ N0 ; λ ∈ C), where 2 F1 (a, b; c; z) denotes the Gauss hypergeometric function defined by 1.5(4). (See Luo and Srivastava [791, p. 5708, Theorem 1]) (α) 83. For the Apostol-Genocchi polynomials Gn (x; λ) of order l (l ∈ N0 ) defined by 1.8(58), derive the following relationship: Gn(α) (x + y; λ) =

n X k=0

α, λ ∈ C; n ∈ N0

 h i n 2 (α−1) (α) (k + 1)Gk (y; λ) − Gk+1 (y; λ) Bn−k (x; λ) k+1 k




with the Apostol-Bernoulli polynomials Bn−k (x; λ) defined by 1.8(1). (See Luo and Srivastava [791, p. 5710, Theorem 2]) 84. For the Apostol-Genocchi polynomials Gn (x; λ), defined by 1.8(58) with α = 1, show that the following integral representation holds true: Gn (z; e

2πiξ

) = 2ne

−2πizξ

Z∞ 0

 M(n; z, t) cosh (2πξ t) + i N(n; x, t) sinh (2πξ t) n−1 t dt cosh (2π t) − cos (2π z)   1 n ∈ N; 0 5 <(z) 5 1; |ξ | < (ξ ∈ R) , 2

where h   nπ i nπ  M(n; z, t) = eπ t cos π z − − e−πt cos πz + 2 2 and h   nπ i nπ  N(n; z, t) = eπ t sin πz − + e−πt sin πz + . 2 2 (See Luo and Srivastava [791, p. 5724, Theorem 19]) 85. Let the so-called λ-Stirling numbers S(n, k; λ) of the second kind be defined by means of the following generating function: ∞

(λez − 1)k X zn = S(n, k; λ) k! n! n=0

(k ∈ N0 ; λ ∈ C),

140

Zeta and q-Zeta Functions and Associated Series and Integrals

so that, obviously, S(n, k) := S(n, k; 1) for the Stirling numbers S(n, k) of the second kind, defined by 1.6(15). Show that each of the following results holds true: ∞   X x λ x = k! S(n, k; λ) k x n

(k ∈ N0 ; λ ∈ C),

k=0

S(n, k; λ) =

  k 1X k jn (−1)k−j λj k! j

(n, k ∈ N0 ; λ ∈ C),

j=0

  k 1X j k (−1) λk−j (k − j)n S(n, k; λ) = k! j

(n, k ∈ N0 ; λ ∈ C),

j=0

S(n, k; λ) = S(n − 1, k − 1; λ) + k S(n − 1, k; λ)

(n, k ∈ N)

and S(n, k; λ) =

 n−1  X n − 1 n−j−1 λ S( j, k − 1; λ) j

(n, k ∈ N).

j=0

(See Luo and Srivastava [791, p. 5716, Theorems 9 to 11]) 86. For the λ-Stirling numbers S(n, k; λ) of the second kind, defined by means of the generating function in Problem 84 above, show that each of the following explicit relationships holds true: S(n, k; λ) = n!

∞   X j (log λ) j−n j=n

n

j!

S( j, k)

(n, k ∈ N0 ; λ ∈ C)

and S(n, k; λ) =

k X λ j (λ − 1)k−j j=0

(k − j)!

S(n, j)

(n, k ∈ N0 ; λ ∈ C),

with the Stirling numbers S(n, k) of the second kind, defined by 1.6(15). (See Luo and Srivastava [791, pp. 5716–5717, Theorem 12])

2 The Zeta and Related Functions This chapter aims at providing a self-contained theory of the Zeta and related functions, which will be required in each of the next chapters. We first introduce (and investigate the various properties and relationships satisfied by) the multiple Hurwitz Zeta function ζn (s, a) (n ∈ N) and consider its relatively more familiar special case when n = 1, that is, the Hurwitz (or generalized) Zeta function ζ (s, a). We then deal rather systematically with the Riemann Zeta function, which (for the main purpose of this book) happens to be the most important member of the significantly large family of Zeta functions considered in this chapter. Other functions (introduced in this chapter) include the Polylogarithm functions, Legendre’s Chi function, Clausen’s integral (or Clausen’s function), the Hurwitz–Lerch Zeta function and so on.

2.1 Multiple Hurwitz Zeta Functions Barnes [97] introduced and studied the generalized multiple Hurwitz Zeta function ζn (s, a | w1 , . . . , wn ) defined for <(s) > n by the n-ple series ζn (s, a | w1 , . . . , wn ) :=

∞ X m1 , ..., mn =0

1 (a + )s

(<(s) > n; n ∈ N),

(1)

where  = m1 w1 + · · · + mn wn and the general conditions for a and the parameters w1 , . . . , wn are given in Barnes [97], who used it in the study of the multiple Gamma functions (see Section 1.4). In this section, we consider only the simple case of (1), when wj = 1 ( j = 1, . . . , n; j, n ∈ N) and ζn (s, a) :=

∞ X

(a + k1 + · · · + kn )−s

(<(s) > n; a 6∈ Z− 0 ),

(2)

k1 , ..., kn =0

which is also referred to as n-ple (or, simply, multiple) Hurwitz Zeta function. We shall give a rather detailed investigation of the properties and characteristics of the function ζn (s, a) in (2), including its analytic continuation. Zeta and q-Zeta Functions and Associated Series and Integrals. DOI: 10.1016/B978-0-12-385218-2.00002-5 c 2012 Elsevier Inc. All rights reserved.

142

Zeta and q-Zeta Functions and Associated Series and Integrals

The Analytic Continuation of ζn (s, a) In order to give the analytic continuation of ζn (s, a), we first recall a criterion for the convergence of an n-ple series attributed to Eisenstein in the work of Forsyth [457], who gave the proof for the double series. Here, we shall give a proof of the general case: Theorem 2.1 The n-ple series: ∞ X

0

m1 , ..., mn =−∞

 −µ m21 + · · · + m2n

(3)

converges, if µ > 21 n, where the prime denotes the exclusion of the case when mj = 0 ( j = 1, . . . , n). Proof. Let the series be arranged in the partial series: for this purpose, choose integers kj , such that 2kj 5 mj < 2kj +1 (1 5 j 5 nj ; kj ∈ N0 ). Let D := {(m1 , . . . , mn ) | 2kj 5 mj < 2kj +1 ; 1 5 j 5 n} and let Ij := [2kj , 2kj +1 ) (1 5 j 5 n) be half-open intervals. Since 2k+1 − 2k = 2k , the number of integers mj in Ij is 2kj (1 5 j 5 n). So the number of n-tuples in D is 2k1 · · · 2kn = 2k1 +···+kn = 2nx , where x :=

k1 + · · · + kn . n

Note that, for (m1 , . . . , mn ) ∈ D, n X

22kj 5 m21 + · · · + m2n <

j=1

n X

22kj +2

j=1

and, by comparing the arithmetic and geometric means of 22kj (1 5 j 5 n), we also have n

1 X 2kj 2 5 2 5 m21 + · · · + m2n . n 2x

j=1

The Zeta and Related Functions

143

Now, for any (m1 , . . . , mn ) in D and any positive real µ, we have 1 (m21 + · · · + m2n )µ

5

1 . 22xµ

Then, the sum of all terms as 1/(m21 + · · · + m2n )µ in D is less than or equal to 2nx 1 1 1 = x(2µ−n) = 2µ−n · · · 2µ−n . 2xµ k 2 2 2 n 1 2 n kn Let k = max{kj | 1 5 j 5 n}. We find that the sum of all the partial series is less than or equal to  n k k k X X X 1 1 1  ··· · · · 2µ−n =  2µ−n 2µ−n n k1 n kn n k1 2 2 2 k1 =0 kn =0 k1 =0  n (4) −(k+1) 2µ−n n 1 − 2  . = 1 1 − 2µ−n 2

n

Taking the limit in (4) as k → ∞, we observe that the n-ple series in (4) converges, if 2µ − n > 0, that is, if µ > 21 n and the sum is n  2µ−n 1   = 2 n . 2µ−n 1 1 − 2µ−n 2 n −1 n 2

This completes the proof of Theorem 2.1.



Let s = σ + it (σ, t ∈ R). First, for convergence, we consider ζn (s, a) in (2) for the case when a > 0: ζn (s, a) =

∞ X

(a + k1 + · · · + kn )−s

(<(s) = σ > n; a > 0).

(5)

k1 , ..., kn =0

Theorem 2.2 The series for ζn (s, a) in (5) converges absolutely for σ > n. The convergence is uniform in every half-plane σ ≥ n + δ (δ > 0), so ζn (s, a) is an analytic function of s in the half-plane σ > n. Proof. Observe that, for σ > 0, ∞ X

0

(k1 + · · · + kn )

−σ

=

k1 , ..., kn =0

5

∞ X

0

[(k1 + · · · + kn )2 ]− 2 σ

k1 , ..., kn =0 ∞ X

0

(k12 + · · · + kn2 )− 2 σ ,

k1 , ..., kn =0

1

1

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Zeta and q-Zeta Functions and Associated Series and Integrals

in which the prime denotes the exclusion of the case when kj = 0 (1 5 j 5 n) and that the last series converges for σ > n by Eisentein’s theorem (Theorem 2.1). Thus, all statements in Theorem 2.2 follow from the inequality: ∞ X

|(a + k1 + · · · + kn )−s | =

k1 , ..., kn =0

5

∞ X

(a + k1 + · · · + kn )−σ

k1 , ..., kn =0 ∞ X

(a + k1 + · · · + kn )−n−δ .

k1 , ..., kn =0

We now choose to recall a convergence theorem concerning term-by-term integration of monotonic sequences of functions, which is due to Le´ vi (see Apostol [64, p. 268, Theorem 10.25]).  Theorem 2.3 Let L(I) denote the set of all Lebesgue-integrable functions on an interval I. Also, let {gn } be a sequence of functions in L(I), such that (a) each gn is P nonnegative almost everywhere on I R g converges. (b) the series ∞ n n=1 I

Then, the series L(I). Moreover, Z g= I

Z X ∞

P∞

n=1 gn

gn =

I n=1

converges almost everywhere on I to a sum function g in

∞ Z X

gn .

n=1 I

Next, we present an integral representation of ζn (s, a), which is given by Theorem 2.4 If <(s) = σ > n, then 0(s) ζn (s, a) =

Z∞ 0

xs−1 e−ax dx (1 − e−x )n

(<(s) > n; n ∈ N).

(6)

Proof. It follows from 1.1(1) that, for <(s) = σ > 0, 0(s) =

Z∞ xs−1 e−x dx. 0

We first keep s real, s > n (n ∈ N), and then extend the result to complex s by analytic continuation. In this Eulerian integral for 0(s), we set x = (a + k1 + · · · + kn )t

(kj ∈ N0 ; j = 1, . . . , n),

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145

and we find that Z∞ 0(s) = (a + k1 + · · · + kn ) e−(a+k1 +···+kn )t t s−1 dt, s

0

so that (a + k1 + · · · + kn )

−s

Z∞ 0(s) = e−(k1 +···+kn )t e−at t s−1 dt. 0

Summing over all kj ∈ N0 (1 5 j 5 n), we obtain ∞ X

ζn (s, a) 0(s) =

Z∞ e−(k1 +···+kn )t e−at t s−1 dt,

k1 , ..., kn =0 0

the series on the right being convergent, if s > n. Now, we wish to interchange the order of summation and integration. The simplest way to justify this process is to regard the integral as a Lebesgue integral. Since the integrand is nonnegative, Le´ vi’s convergence theorem (Theorem 2.3) implies that the series: ∞ X

Z∞ e−(a+k1 +···+kn )t e−at t s−1 dt

k1 , ..., kn =0 0

converges almost everywhere to a sum function, which itself is Lebesgue-integrable on [0, ∞) and that ζn (s, a) 0(s) =

Z∞

∞ X

e−(k1 +···+kn )t e−at t s−1 dt.

0 k1 , ..., kn =0

But, if t > 0, we have 0 < e−t < 1, and, hence, ∞ X

e−kt =

k=0

1 , 1 − e−t

the series being a geometric series. We, therefore, have ∞ X k1 , ..., kn =0

e−(k1 +···+kn )t e−at t s−1 =

e−at t s−1 (1 − e−t )n

146

Zeta and q-Zeta Functions and Associated Series and Integrals

almost everywhere on [0, ∞); in fact, everywhere except at 0, so we get ζn (s, a) 0(s) =

Z∞ 0

e−at t s−1 dt. (1 − e−t )n

This proves (6) for real s and s > n (n ∈ N). To extend this integral representation to all complex s = σ + it with σ > n, we note that both functions on the left-hand side of (6) are analytic for σ > n. To show that the right member is also analytic, we assume that n + δ 5 σ 5 c, where c > n and δ > 0, and write Z∞ −at σ −1 Z∞ −at s−1 e t e t (1 − e−t )n dt = (1 − e−t )n dt 0

0

 1 ∞ Z Z e−at tσ −1  = dt. +  (1 − e−t )n 0

1

If 0 5 t 5 1, we have tσ −n 5 tδ , and, if t = 1, we have tσ −n 5 tc−n . Also, since et − 1 = t for t = 0, we have Z1 0

e−at tσ −1 dt 5 (1 − e−t )n

5

Z1 0

e(n−a)t tδ+n−1 dt (et − 1)n

 R n−a 1 tδ−1 dt =  e 0  R 1 0

tδ−1 dt =

1 δ

en−a δ

(0 < a 5 n) (a > n)

and Z∞ 1

e−at tσ −1 dt 5 (1 − e−t )n

Z∞ 0

e−at tc−1 dt = 0(c) ζn (c, a). (1 − e−t )n

This shows that the integral in (6) converges uniformly in every strip n + δ 5 σ 5 c, where δ > 0 and, therefore, represents an analytic function in every such strip; hence, also in the half-plane σ > n. Thus, by the principle of analytic continuation, (6) holds true for all s with <(s) = σ > n (n ∈ N).  In order to extend ζn (s, a) to the half-plane on the left of the line σ = n, we derive another representation in terms of a contour integral. The contour C is essentially a Hankel’s loop (cf., e.g., Whittaker and Watson [1225, p. 245]), which starts from ∞

The Zeta and Related Functions

147

along the upper side of the positive real axis, encircles the origin once in the positive (counter-clockwise) direction and then returns to ∞ along the lower side of the positive real axis. The loop C is composed of three parts C1 , C2 and C3 , where C2 is a positivelyoriented circle of radius c < 2π about the origin and C1 and C3 are the upper and lower edges of a cut in the complex z-plane along the positive real axis, traversed as described above. Thus, we can use the parameterizations −z = re−π i on C1 and −z = reπi on C3 , where r varies from c to ∞. Theorem 2.5 If a > 0, then the function defined by the following contour integral: 1 In (s, a) = − 2π i

(0+) Z ∞

(−z)s−1 e−az dz (1 − e−z )n

(7)

is an entire function of s. Moreover, ζn (s, a) = 0(1 − s) In (s, a)

(<(s) = σ > n).

(8)

Proof. Here (−z)s means rs e−π is on C1 and rs eπis on C3 . We consider an arbitrary compact disk |s| 5 M and prove that the integrals along C1 and C3 converge uniformly on every such disk. Since the integrand in (7) is an entire function of s, this will prove that In (s, a) is also an entire function of s for a > 0. Along C1 , we have, for r = 1, |(−z)s−1 | = rσ −1 e−π i(σ −1+it) = rσ −1 eπ t 5 rM−1 eπ M , since |s| 5 M. Similarly, along C3 , we have, for r = 1, (−z)s−1 = rσ −1 |eπ i (σ −1+it) | = rσ −1 e−π t 5 rM−1 eπ M . Hence, on either C1 or C3 , we have, for r = 1, (−z)s−1 e−az rM−1 eπ M e−ar eπ M M−1 −ar ·e . (1 − e−z )n 5 (1 − e−r )n 5 1 − e−1
n ·r But the integral: Z∞

rM−1 e−ar dr

c

converges, if c > 0; this shows that the integrals along C1 and C3 converge uniformly on every compact disk |s| 5 M, and, hence, In (s, a) is an entire function of s. To prove (8), we write   Z Z Z   −2πi In (s, a) =  + +  (−z)s−1 g(−z) dz, C1

C2

C3

148

Zeta and q-Zeta Functions and Associated Series and Integrals

where g(−z) =

e−az . (1 − e−z )n

On C1 and C3 , we have g(−z) = g(−r), and, on C2 , we write −z = ceiθ , where θ varies from 2π to 0. This gives us Zc

−2πi In (s, a) =

r

s−1 −π i (s−1)

e

Z0 g(−r) dr − i

∞

cs−1 e(s−1)iθ ceiθ g(ceiθ ) dθ

2π

Z∞ + rs−1 eπi (s−1) g(−r) dr

(9)

c

Z∞ Z0 s−1 s = −2i sin(πs) r g(−r) dr − ic eisθ g(ceiθ ) dθ. c

2π

Dividing both sides of (9) by −2i, we obtain π In (s, a) = sin(πs)I1 (s, c) + I2 (s, c), where Z∞ I1 (s, c) = rs−1 g(−r)dr

cs and I2 (s, c) = 2

c

Z0

eisθ g(ceiθ )dθ.

2π

Now, let c → 0. We find that, in view of (6), lim I1 (s, c) =

Z∞

c→0

0

rs−1 e−ar dr = 0(s) ζn (s, a) (1 − e−r )n

(σ = <(s) > n).

We next show that lim I2 (s, c) = 0.

c→0

To do this, we note that g(−z) is analytic in |z| < 2π, except for a pole of order n at z = 0. Therefore, zn g(−z) is analytic everywhere inside |z| < 2π and, hence, is bounded there, that is, |g(−z)| 5

A |z|n

(|z| = c < 2π),

The Zeta and Related Functions

149

and A is a positive constant. We, thus, have cσ |I2 (s, c)| 5 2

Z2π A e−tθ · n dθ 5 πA e2π |t| cσ −n . c 0

If σ > n and c → 0, we find that I2 (s, c) → 0; hence πIn (s, a) = sin(πs) 0(s) ζn (s, a),

(10)

which is, in terms of 1.1(12), seen to be equivalent to (8).



In the equation (8), valid for σ > n, the function In (s, a) is an entire function of s, and 0(1 − s) is analytic for every complex s for s ∈ C \ N. We, therefore, can use this equation to define ζn (s, a) for σ 5 n, that is, outside σ > n as desired. Definition 2.1 If <(s) = σ 5 n, we define ζn (s, a) by ζn (s, a) := 0(1 − s) In (s, a),

(11)

where In (s, a) is given in (7). This equation (11) provides the analytic continuation of ζn (s, a) to the whole complex s-plane. Theorem 2.6 The function ζn (s, a), defined by (11), is analytic for all s except for simple poles at s = k (1 5 k 5 n), with their respective residues given by Res ζn (s, a) = s=k

1 dn−k zn e−az lim n−k (n − k)!(k − 1)! z→0 dz (1 − e−z )n

(k = 1, . . . , n; n ∈ N). (12)

In particular, when s = n, its residue is 1/(n − 1)!. Proof. Since In (s, a) is an entire function of s for a > 0, the only possible singularities of ζn (s, a) are the poles of 0(1 − s). Since 1/ 0(1 − s) has simple zeros at s ∈ N, 0(1 − s) has simple poles at s ∈ N. But Theorem 2.2 shows that ζn (s, a) is analytic in <(s) = σ > n, and so s = 1, . . . , n are the only poles of ζn (s, a). We next prove the assertion (12). If s is any integer, say s = k, the integrand in the contour integral for In (s, a) in (7) takes the same values on both C1 and C3 , and, hence, the integrals along C1 and C3 cancel, giving us 1 In (k, a) = − 2πi

Z C2

(−z)k−1 e−az
n dz 1 − e−z

= − Res f (z), z=0

150

Zeta and q-Zeta Functions and Associated Series and Integrals

where, for convenience,

f (z) :=

(−z)k−1 e−az . (1 − e−z )n

We observe that, for 1 5 k 5 n, the function f (z) has a pole of order n + 1 − k at z = 0. We, therefore, have

In (k, a) =

(−1)k dn−k zn e−az lim n−k . (n − k)! z→0 dz (1 − e−z )n

(13)

To find the residue of ζn (s, a) at s = k (1 5 k 5 n), by using 1.1(12), we compute the limit: lim (s − k)ζn (s, a) = lim (s − k)0(1 − s)In (s, a)

s→k

s→k

=

π In (k, a) s−k · lim 0(k) s→k sin(πs)

=

In (k, a) , (−1)k (k − 1)!

which, by virtue of (13), immediately yields (12).



Relationship between ζn (s, x) and B(α) n (x) By using the multiple Hurwitz Zeta function, Choi [262] derived the following explicit (α) formula for Bn (x) different from 1.7(28):   n−1 n−1   X n+k X ` j Bk+j+1 (x) (−1) s(n, ` + 1)x`−j n k+j+1 j

(n) Bn+k (x) = n

j=0

(14)

`=j

in terms of the Stirling numbers s(n, k) of the first kind (see Section 1.6). The value of ζn (−`, x) can be calculated explicitly for ` ∈ N0 . Taking s = −` in the relation (8), with a replaced by x, we find that ζn (−`, x) = 0(1 + `)In (−`, x) = `! In (−`, x).

(15)

The Zeta and Related Functions

151 (α)

Now, from the definition (19) of the generalized Bernoulli polynomials Bn (x), we have In (−`, x) = −

1 2πi

(−z)−`−1 e−xz
n dz 1 − e−z

Z C

(−z)−`−1 e−xz
n z=0 1 − e−z

= − Res

zn e(n−x)z z=0 (ez − 1)n ∞ X zk (n) = (−1)` Res z−n−`−1 Bk (n − x) z=0 k!

= (−1)` Res z−n−`−1

k=0

(n)

B (n − x) = (−1)` n+` , (n + `)! which, in view of 1.7(20) and (15), yields the desired relationship: ζn (−`, x) = (−1)n

`! (n) B (x) (n + `)! n+`

(` ∈ N0 ).

(16)

Setting n = 1 in (16), we have the well-known result: ζ (−`, x) = −

B`+1 (x) `+1

(` ∈ N0 ),

(17)

where ζ (s, x) := ζ1 (s, x) is the Hurwitz (or generalized) Zeta function (see Section 2.2). The number of solutions of k1 + · · · + kn = k

(k ∈ N0 ; (k1 , . . . , kn ) ∈ N0 n )

is equal to the coefficient of xk in the Maclaurin series expansion of (1 − x)−n : (1 − x)

−n

  ∞  ∞  X X −n k+n−1 k k = (−x) = x . k n−1 k=0

(18)

k=0

The multiple Hurwitz Zeta function in (5) can, thus, be expressed as a simple series: ζn (s, x) =

 ∞  X k+n−1 (x + k)−s . n−1 k=0

(19)

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Zeta and q-Zeta Functions and Associated Series and Integrals

From 1.6(1) or 1.6(5), we find that 

 n−1 X k+n−1 1 |s|(n, j + 1)kj , = (n − 1)! n−1

(20)

j=0

where |s|(n, k) := (−1)n+k s(n, k) are often called the unsigned or absolute Stirling numbers of the first kind. Combining (19) and (20), we have   ∞ n−1 X X 1  |s|(n, j + 1)kj  (x + k)−s , ζn (s, x) = (n − 1)! j=0

k=0

which, by virtue of the identity: `   X ` k = {(−x) + (x + k)} = (−x)`−j (x + k)j , j `

`

j=0

yields    ! n−1 `   ∞ X  X X 1 ` 1  |s|(n, ` + 1) ζn (s, x) = (−x)`−j    (n − 1)! (x + k)s−j j `=0

=

1 (n − 1)!

n−1 X

|s|(n, ` + 1)

`=0

j=0

`  X j=0

k=0

 ` (−x)`−j ζ (s − j, x). j

Next, it is easy to show that ζn (s, x) is expressible as a finite combination of the generalized Zeta function ζ (s, x) with polynomial coefficients in x: ζn (s, x) =

n−1 X

pn,j (x)ζ (s − j, x),

(21)

j=0

where pn,j (x) =

  n−1 X ` 1 (−1)n+1−j s(n, ` + 1)x`−j . (n − 1)! j

(22)

`=j

We shall now find pn,j (x) in (22) as a polynomial in x of degree n − 1 − j with rational coefficients. Since ζ (s, x) can be continued analytically to a meromorphic function, having a simple pole at s = 1 with its residue 1, the representation (21) shows that ζn (s, x) is

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153

analytic for all s, except for simple poles only at s = k (k = 1, . . . , n; n ∈ N) with their respective residues given by Res ζn (s, x) = pn,k−1 (x) s=k

(k = 1, . . . , n; n ∈ N).

(23)

In view of 1.6(3) and 1.6(6), ζn (s, x) can be expressed explicitly for the first few values of n: ζ2 (s, x) = (1 − x)ζ (s, x) + ζ (s − 1, x),    3 1 1 2 x − 3x + 2 ζ (s, x) + − x ζ (s − 1, x) + ζ (s − 2, x), ζ3 (s, x) = 2 2 2    1 n 3 − x + 6x2 − 11x + 6 ζ (s, x) + 3x2 − 12x + 11 ζ (s − 1, x) ζ4 (s, x) = 6 − (3x − 6)ζ (s − 2, x) + ζ (s − 3, x)}.

(24)

Letting s = −` in (21) and applying (16), we obtain the desired formula (14). Also, upon setting x = n (n ∈ N) in (14) and making use of 1.7(21), we find that (n) Bn+k

= (−1)

n+k

  n−1 n−1   X ` n+k X j Bk+j+1 (n) s(n, ` + 1)n`−j . n (−1) k+j+1 j n

(25)

`=j

j=0

Now, we express Ress=k ζn (s, x) in (12) in a more recognizable form: (n)

Res ζn (s, x) = s=k

Bn−k (n − x) (n − k)!(k − 1)!

.

(26)

Indeed, it follows from 1.7(19) and (12) that ∞

lim (s − k)ζn (s, x) =

s→k

1 zj dn−k X (n) lim n−k Bj (n − x) (n − k)!(k − 1)! z→0 dz j! j=0

=

(n) Bn−k (n − x)

(n − k)!(k − 1)!

.

The Vardi-Barnes Multiple Gamma Functions Vardi [1190, p. 498] gave another expression for the multiple Gamma functions 0n (a) (see Section 1.4), whose general form was also studied by Barnes [97]: " 0n (a) =

n Y

m=1

(−1)m ( a ) Rn−m+1m−1

# Gn (a)

(n ∈ N),

(27)

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Zeta and q-Zeta Functions and Associated Series and Integrals

where   Gn (a) := exp ζn0 (0, a)

with ζn0 (s, a) =

∂ ζn (s, a) ∂s

and m X

Rm := exp

! ζk0 (0, 1)

with R0 = 1.

(28)

k=1

In particular, the special cases of (27), when n = 1 and n = 2, give other forms of the simple and double Gamma functions 01 = 0 and 02 :   0(a) = exp −ζ 0 (0) + ζ 0 (0, a) √   = 2π exp ζ 0 (0, a) ,

(29)

where ζ (s) := ζ (s, 1) is the Riemann Zeta function (see Section 2.3);   1 1 1 02 (a) = A(2π) 2 − 2 a exp − + ζ20 (0, a) , 12

(30)

where we have used (24) and the known identity (see Voros [1201, p. 462, Eq. (A.11)]): 1 − ζ 0 (−1). 12

log A =

(31)

Here we can give another proof of the multiplication formula for 02 (see 1.4(21)) different from that of Barnes [94], by using (30) (see Choi and Quine [278]). We consider n−1 X n−1 X `=0 j=0

`=0 j=0 k1 , k2 =0

n−1 X n−1 X

∞ X

= ns = ns

  X  −s n−1 X n−1 X ∞ `+j `+j ζ2 s, a + + k1 + k2 = a+ n n (na + ` + j + nk1 + nk2 )−s

k1 , k2 =0 `=0 j=0 ∞ X

(na + k1 + k2 )−s = ns ζ2 (s, na),

k1 , k2 =0

which, upon differentiating with respect to s, yields n−1 X n−1 X `=0 j=0

ζ20



`+j s, a + n



= (log n) ns ζ2 (s, na) + ns ζ20 (s, na).

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155

By virtue of (30), we readily obtain the following multiplication formula for 02 : n−1 Y n−1 Y

02



`=0 j=0

`+j a+ n



1

= C (n) (2π)− 2 n(n−1)a n−

n2 a2 2 +na

02 (na),

(32)

where 2 −1

C (n) := An

·e 12 (1−n ) ·(2π) 2 (n−1) ·n 12 . 1

1

2

5

An interesting identity is also obtained from (32): n−1 Y n−1 Y

0

02

`=0 j=0



`+j n

 =

C (n) , n

(33)

where the prime denotes the exclusion of the case when ` = 0 = j.

2.2 The Hurwitz (or Generalized) Zeta Function The Hurwitz (or generalized) Zeta function ζ (s, a) is defined by ζ (s, a) :=

∞ X

(<(s) > 1; a 6∈ Z− 0 ).

(k + a)−s

(1)

k=0

It is easy to see that ζ (s, a) = ζ1 (s, a) for the case when n = 1 in 2.1(2). Thus, we can deduce many properties of ζ (s, a) from those of ζn (s, a) in Section 2.1. Indeed, the series for ζ (s, a) in (1) converges absolutely for <(s) = σ > 1. The convergence is uniform in every half-plane σ ≥ 1 + δ (δ > 0), so ζ (s, a) is an analytic function of s in the half-plane <(s) = σ > 1. Setting n = 1 in 2.1(6), we have the integral representation: 0(s) ζ (s, a) =

Z∞ 0

Z1 = 0

xs−1 e−ax dx = 1 − e−x xa−1 1−x

 log

1 x

Z∞ 0

xs−1 e−(a−1)x dx ex − 1 (2)

s−1 dx

(<(s) > 1; <(a) > 0).

Moreover, ζ (s, a) can be continued meromorphically to the whole complex s-plane (except for a simple pole at s = 1 with its residue 1) by means of the contour integral

156

Zeta and q-Zeta Functions and Associated Series and Integrals

representation (see Theorem 2.5): 0(1 − s) ζ (s, a) = − 2πi

Z C

(−z)s−1 e−az dz, 1 − e−z

(3)

where the contour C is the Hankel loop of Theorem 2.5. The connection between ζ (s, a) and the Bernoulli polynomials Bn (x) is also given in 2.1(17). From the definition (1) of ζ (s, a), it easily follows that ζ (s, a) = ζ (s, n + a) +

n−1 X

(k + a)−s

(n ∈ N);

(4)

k=0

    ∞ X 1 1 1 ζ s, a − ζ s, a + = 2s (−1)n (a + n)−s . 2 2 2

(5)

n=0

Hurwitz’s Formula for ζ (s, a) The series expression ζ (s, a) was originally meaningful for σ > 1 (s = σ + it). Hurwitz obtained another series representation for ζ (s, a) valid in the half-plane σ < 0: o 1 0(s) n − 1 πis e 2 L(a, s) + e 2 π is L(−a, s) s (2π) (0 < a 5 1, σ = <(s) > 1; 0 < a < 1, σ > 0), ζ (1 − s, a) =

(6)

where the function L(x, s) is defined by

L(x, s) :=

∞ 2π inx X e ns

(x ∈ R; σ = <(s) > 1),

(7)

n=1

which is often referred to as the periodic (or Lerch) Zeta function. We note that the Dirichlet series in (7) is a periodic function of x with period 1 and that L(1, s) = ζ (s), the Riemann Zeta function (see Section 2.3). The series in (7) converges absolutely for σ > 1. Yet, if x ∈ / Z, the series can also be seen to converge conditionally for σ > 0. So, the formula (6) is also valid for σ > 0, if a 6= 1. We observe that the function L(x, s) in (7) is a linear combination of the Hurwitz Zeta functions, when x is a rational number. Indeed, setting x = p/q (1 5 p 5 q; p, q ∈ N) in (7), the terms in (7) can be rearranged according to the residue classes mod q, by letting n = kq + r (1 5 r 5 q; k ∈ N0 ),

The Zeta and Related Functions

157

which gives us, for σ > 1,     X  X q X ∞ ∞ exp 2π irp q 2πinp p 1 exp ,s = = L q ns q (kq + r)s n=1 r=1 k=0   ∞ q 1 X 2πirp X 1  s = s exp r q q r=1 k=0 k + q     q r 2πirp 1 X ζ s, . exp = s q q q 

r=1

Therefore, if we take a = p/q in the Hurwitz formula (6), we obtain: 

p ζ 1 − s, q



    q 2 0(s) X πs 2πrp r = cos − ζ s, (2πq)s 2 q q r=1

(8)

(1 5 p 5 q; p, q ∈ N), which holds true, by the principle of analytic continuation, for all admissible values of s ∈ C.

Hermite’s Formula for ζ (s, a) We, first, recall Plana’s summation formula: n X k=m

1 f (k) = [ f (m) + f (n)] + 2

Zn

f (τ ) dτ − 2

m

Z∞ 0

q(m, t) − q(n, t) dt, e2πt − 1

(9)

where f (z) is a bounded analytic function in m 5 <(z) 5 n and q(λ, t) =

√ 1 [ f (λ + it) − f (λ − it)] (i = −1). 2i

Choose the function f (z) in (9) as follows: f (z) = (a + z)−s

(s = σ + it; | arg(a + z)| < π).

Then, it is easy to see that    h i− 1 s y 2 q(x, y) = − (a + x)2 + y2 sin s arctan . x+a In view of the elementary inequality:     y arctan 5 min π , |y| , x+a 2 x+a

(10)

158

Zeta and q-Zeta Functions and Associated Series and Integrals

we obtain   h i1−1σ 1 2 2 2 2 −1 π|s| (|y| > a); |q(x, y)| 5 (a + x) + y y sinh 2   h i− 1 σ 2 sinh y|s| (|y| < a). |q(x, y)| 5 (a + x)2 + y2 x+a

(11)

Making use of (11), it is easily seen that the integral: Z∞

 −1 q(x, y) e2π y − 1 dy (σ > 0)

0

converges when x = 0 and tends to 0 as x → ∞. Also, the improper integral: Z∞ (a + x)−s dx

(σ = <(s) > 1)

0

converges. Therefore, if σ > 1, it is valid to make n → ∞ (m = 0) in (9) with the function f (z) in (10). Thus, we readily obtain Hermite’s formula for ζ (s, a): 1 a1−s ζ (s, a) = a−s + +2 2 s−1

Z∞  − 1 s n  y o dy 2 a2 + y2 sin s arctan . a e2π y − 1

(12)

0

We note that the integral involved in (12) converges for all admissible values of s ∈ C. Moreover, the integral is an entire function of s. A special case of the formula (12) when a = 1 is attributed to Jensen. Setting s = 0 in (12), we have ζ (0, a) =

1 − a, 2

(13)

which is also obtained from 2.1(17) in view of 1.7(8). If we set z = s in 1.3(30) and differentiate the resulting equation with respect to s, we find that 1 ψ(s) = log s − − 2 2s

Z∞ 0

t 2 + s2

t dt

e2π t − 1

(<(s) > 0).

(14)

Taking the limit in (12) as s → 1, by virtue of the uniform convergence of the integral in (12), we get  Z∞ 1 a1−s − 1 1 y dy

, lim ζ (s, a) − = lim + +2 2 2 s→1 s→1 s − 1 s−1 2a a + y e2π y − 1 

0

The Zeta and Related Functions

159

which, in view of (14), yields  0 0 (a) 1 =− = −ψ(a). lim ζ (s, a) − s→1 s−1 0(a) 

(15)

Differentiating (12) with respect to s and setting s = 0 in the resulting equation, we have 


   Z∞ arctan ay 1 d ζ (s, a) log a − a + 2 = a− dy, ds 2 e2π y − 1 s=0

(16)

0

which, by virtue of 1.2(30), yields d 1 ζ (s, a) = log 0(a) − log (2π), ds 2 s=0

(17)

which is equivalent to the identity 2.1(29). In addition to (17), it is easy to find from the definition (1) of ζ (s, a) that ∂ ζ (s, a) = −s ζ (s + 1, a). ∂a

(18)

The respective special cases of (15) and (17) when a = 1, by means of 1.2(4) and 1.1(13), become 

   1 1 lim ζ (s, a) − = lim ζ (1 + , a) − =γ s→1 →0 s−1 

(19)

1 ζ 0 (0) = − log (2π), 2

(20)

and

where ζ (s) is the Riemann Zeta function (see Section 2.3).

Further Integral Representations for ζ (s, a) In addition to (12), some known integral representations of ζ (s, a) are recalled here: 0(s) ζ (s, a) =

Z∞ 0

Z1 = 0

t s−1 e−at dt = 1 − e−t ta−1 1−t

Z∞ 0

t s−1 e−(a−1)t dt et − 1

  1 s−1 log dt t

(21)
<(s) > 1; <(a) > 0 ,

160

Zeta and q-Zeta Functions and Associated Series and Integrals

which is the same as (2); 1 a1−s 1 ζ (s, a) = a−s − + 2 1 − s 0(s)

Z∞  0

1 1 1 − + t e −1 t 2



e−at t s−1 dt

(22)

(<(s) > −1; <(a) > 0); h  i t Z∞ h 1 i s−2 cos (s − 1) arctan (1−s) 2a−1 π2 2   ζ (s, a) = t2 + (2a − 1)2 dt 2 1 s−1 cosh πt (23) 2 0   1 ; <(a) > 2   Z∞ 1 t−s ζ (s, a) = cos π s sin(2πa) dt 2 cosh(2πt) − cosh(2πa) 0

  Z∞ −s  t cosh(2πa) − e−2π t 1 πs dt + sin 2 cosh(2πt) − cosh(2πa) 

(24)

0

(<(s) < 1 when 0 < <(a) < 1; <(s) < 0 when a = 1); n X 0(k + s − 1) Bk −k−s+1 a 0(s) k! k=0 ! Z∞ n X 1 1 Bk k−1 −at s−1 + − t e t dt 0(s) et − 1 k!

ζ (s, a) = a−s +

0

(25)

k=0

(<(s) > −(2n − 1); <(a) > 0; n ∈ N0 ).

Some Applications of the Derivative Formula (17) We begin by giving another proof of the derivative formula (17). Indeed, by the analytic continuation of ζ (s, a) and the special case of (4) when n = 1, we observe that ζ (s, a + 1) = ζ (s, a) − a−s ,

(26)

which, upon differentiating with respect to s and setting s = 0 in the resulting equation, yields ζ 0 (0, a + 1) = ζ 0 (0, a) + log a. Let f (a) := exp[ζ 0 (0, a)]. We then have f (a + 1) = a f (a) (a > 0) and ∞ X d2 d2 d 1 log f (a) = ζ (s, a) = > 0 (a > 0), 2 2 da da ds (k + a)2 s=0 k=0

The Zeta and Related Functions

161

which implies that f (a) is logarithmically convex on (0, ∞). Thus, by appealing to the Bohr-Mollerup theorem (Theorem 1.1), we obtain, for some constant C, f (a) = C 0(a), which, for a = 1, yields 1

C = exp[ζ 0 (0)] = (2π)− 2 . This completes our second proof of the derivative formula (17). Many authors gave seemingly different proofs of Stirling’s formula 1.1(52) (see e.g., Blyth and Pathak [135], Choi [261], Diaconis and Freeman [380] and Patin [889]). Here, by taking the limit in (16) as a → ∞, we have  lim

a→∞

 1 ζ (0, a) + a + log a − a log a = 0, 2 0

(27)

which, upon taking the exponential and using (17), immediately yields Stirling’s formula 1.1(33). Combining formulas (17) and 1.1(42), we obtain a formula for the Beta function B(α, β) :   1 B(α, β) = (2π) 2 exp ζ 0 (0, α) + ζ 0 (0, β) − ζ 0 (0, α + β) , where ζ 0 (s, a) = identities:

(28)

∂ ∂s ζ (s, a). Applying the formula (16) and the following trigonometric

a+b 1 − ab a−b arctan a − arctan b = arctan 1 + ab arctan a + arctan b = arctan

(ab < 1), (29) (ab > −1)

to (28), we can readily deduce an integral representation of B(α, β) (cf. Choi and Nam [276]): α α− 2 β β− 2 1

1

B(α, β) =

α+β− 12

(α + β)

1

(2π) 2 eI(α,β)

(α > 0, β > 0),

where, for convenience, I(α, β) := 2ρ

Z∞

 arctan

0

 (t3 + t)ρ 3 dt 2π tρ αβ(α + β) e −1

  ρ 2 := α 2 + αβ + β 2 .

(30)

162

Zeta and q-Zeta Functions and Associated Series and Integrals

Differentiating both sides of (22) with respect to s and letting s = 0 in the resulting equation, we obtain 

1 ζ (0, a) = a − 2 0

Z∞ 

 log a − a +

0

1 1 1 − + t e −1 t 2



e−at dt t

(31)

(<(a) > 0). In a similar manner, (31) leads us to another integral representation of B(α, β) (cf. Choi and Nam [276]): α α− 2 β β− 2 1

B(α, β) =

1

α+β− 12

(α + β)

1

(2π) 2 eJ(α,β)

(α > 0; β > 0),

(32)

where, for convenience,

J(α, β) :=

Z∞ 0

  1 1  −αt 1 −βt −(α+β)t dt − + e + e − e . et − 1 t 2 t

Another Form for 02 (a) From 2.1(24) and 2.1(30), by virtue of (17), we obtain another form for the double Gamma function 02 (a) :   1 02 (a) = A {0(a)}1−a exp − + ζ 0 (−1, a) 12

(a > 0),

(33)

∂ ζ (s, a). where ζ 0 (s, a) = ∂s In addition to the integral representation 1.4(78), we can express log 02 (a) as improper integrals in many ways. For example, we give two integral representations for log 02 (a):

  1 a2 1 2 1 log 02 (a) = − + log A − + a − a log a + (1 − a) log 0(a) 12 4 2 2 ∞   Z  1 2 2 1 t +2 (a + t ) 2 sin arctan log(a2 + t2 ) (34) 2 a 0    t t dt 2 2 12 + (a + t ) cos arctan arctan (<(a) > 0); 2π a a e t −1

The Zeta and Related Functions

163

 2  a2 a a 1 log 02 (a) = log A − + − + log a + (1 − a) log 0(a) 4 2 2 12  Z∞ 1 1 t e−at 1 − + − dt (<(a) > 0). − et − 1 t 2 12 t2

(35)

0

Indeed, by differentiating Hermite’s formula (12) for ζ (s, a), with respect to s, letting s → −1 in the resulting equation and applying (17) and the identity for ζ 0 (−1, a), we readily obtain (34). Conversely, setting n = 2 in (25), we have Z∞

sa−s−1 a−s a1−s 1 ζ (s, a) = + + + 12 2 s − 1 0(s)

0

 1 1 1 t − + − t s−1 e−at dt et − 1 t 2 12 (36) (<(s) > −3; <(a) > 0).

Employing the same technique as in getting (34), by making use of (36) and considering the following identities: d 1 = −1 ds 0(s) s=−1

and

1 = 0, 0(s) s=−1

we obtain (35). Glaisher [484, p. 47] expressed the Glaisher-Kinkelin constant A given in 1.3(2) as an integral: 1

 7 1 1 2 A = 2 36 π − 6 exp  + 3 3

Z2

  log 0(t + 1) dt.

(37)

0

By setting a = 1 in (34) and (35), we can also obtain integral representations of log A: 1 log A = − 2 3

Z∞

1 1 (1 + t2 ) 2 sin (arctan t) log(1 + t2 ) 2

0 1

+ (1 + t2 ) 2 cos (arctan t) arctan t

o

(38)

dt e2π t − 1

and 1 log A = + 4

Z∞  0

1 1 1 t − + − et − 1 t 2 12



e−t dt. t2

(39)

164

Zeta and q-Zeta Functions and Associated Series and Integrals

The formula (35) can be used to obtain an asymptotic formula for log 02 (a) by first observing that, for some M > 0 and for all a > 0, Z∞  1 1 1 t e−at M dt − + − < a . et − 1 t 2 12 t2 0

Thus, by employing 1.1(34), we have  2  1 a 5 3a2 −a− − −a+ log a log 02 (a) = log A + 4 12 2 12   1 + (1 − a) log(2π) + O a−1 (a → ∞; a > 0), 2

(40)

which may be compared with 1.3(7).

2.3 The Riemann Zeta Function The Riemann Zeta function ζ (s) is defined by P ∞ ∞ P 1 1 1  = (<(s) > 1)  s −s  n (2n−1)s 1−2 n=1 n=1 ζ (s) := ∞ P  (−1)n−1   11−s (<(s) > 0; s 6= 1). ns 1−2

(1)

n=1

It is easy to see from the definitions (1) and 2.2(1) that  
−1 1 ζ s, ζ (s) = ζ (s, 1) = 2s − 1 = 1 + ζ (s, 2) 2

(2)

and  m−1  j 1 X ζ s, ζ (s) = s m −1 m

(m ∈ N \ {1}).

(3)

j=1

In view of (2), we can deduce many properties of ζ (s) from those of ζ (s, a) given in P −s in (1) represents an analytic function Section 2.2. In fact, the series ζ (s) = ∞ n=1 n of s in the half-plane <(s) = σ > 1. Setting a = 1 in 2.2(2), we have an integral representation of ζ (s) in the form: 0(s) ζ (s) =

Z∞ 0

Z1 = 0

xs−1 e−x dx = 1 − e−x 1 1−x

 log

1 x

Z∞ 0

xs−1 dx ex − 1 (4)

s−1 dx

(<(s) > 1).

The Zeta and Related Functions

165

Furthermore, just as ζ (s, a), ζ (s) can be continued meromorphically to the whole complex s-plane (except for a simple pole at s = 1 with its residue 1) by means of the contour integral representation: ζ (s) = −

0(1 − s) 2πi

(−z)s−1 e−z dz, 1 − e−z

Z C

(5)

where the contour C is the Hankel loop of Theorem 2.5. Now 2.2(19) and (5), together, imply that the Laurent series of ζ (s) in a neighborhood of its pole s = 1 has the form: ∞

ζ (s) =

X 1 +γ + an (s − 1)n , s−1

(6)

n=1

where γ is the Euler-Mascheroni constant given in 1.1(3) and an is also expressed as (see Ivic´ [586, pp. 4–6]): ) ( m X (log k)n (log m)n+1 − an = lim m→∞ k n+1

(n ∈ N).

(7)

k=1

The Riemann Zeta function ζ (s) in (1) plays a central roˆ le in the applications of complex analysis to number theory. The number-theoretic properties of ζ (s) are exhibited by the following result, known as Euler’s formula, which gives a relationship between the set of primes and the set of positive integers: ζ (s) =

Y

1 − p−s


−1

(<(s) > 1),

(8)

p

where the product is taken over all primes. From 2.2(4), we have [cf. Equation (2) for the special cases when n = 0 and n = 1]

ζ (s) = ζ (s, n + 1) +

n X

k−s

(n ∈ N0 ).

(9)

k=1

The connection between ζ (s) and the Bernoulli numbers is given as follows:

ζ (−n) =

 1  − 2

(n = 0)

  Bn+1 − n+1

(n ∈ N),

(10)

which is deduced by setting x = 1 in 2.1(17) and using 1.6(5).

166

Zeta and q-Zeta Functions and Associated Series and Integrals

Riemann’s Functional Equation for ζ (s) The special case of 2.2(8), when p = 1 = q, yields Riemann’s functional equation for ζ (s): ζ (1 − s) = 2(2π)

−s

 1 π s ζ (s) 0(s) cos 2 

(11)

or, equivalently, ζ (s) = 2(2π)s−1 0(1 − s) sin



 1 πs ζ (1 − s). 2

Taking s = 2n + 1 (n ∈ N) in (11), the factor cos ζ (−2n) = 0



(12) 

1 2πs

vanishes, and we find that

(n ∈ N),

(13)

which are often referred to as the trivial zeros of ζ (s). The equation (13) can also be proven by combining (10) and 1.7(7). By using the Legendre duplication formulas 1.1(29) and 1.1(12), it is not difficult to see that the functional equations (11) or (12) can be written in a simpler form: 8(s) = 8(1 − s),

(14)

where the function 8(s) is defined by 1

8(s) := π − 2 s 0



 1 s ζ (s). 2

(15)

The function 8(s) has simple poles at s = 0 and s = 1. According to Riemann, to remove these poles, we multiply 8(s) by 12 s(1 − s) and define ξ(s) :=

1 s(1 − s) 8(s), 2

(16)

which is an entire function of s and satisfies the functional equation: ξ(s) = ξ(1 − s).

(17)

Setting s = 2n (n ∈ N) in (11) and applying (10), we have the well-known identity: ζ (2n) = (−1)n+1

(2π)2n B2n 2 (2n)!

(n ∈ N0 ),

(18)

The Zeta and Related Functions

167

which, in view of 1.7(7), enables us to list the following special values: π4 π6 π2 , ζ (4) = , ζ (6) = , 6 90 945 π8 π 10 ζ (8) = , ζ (10) = , ... . 9450 93555

ζ (2) =

(19)

We may recall here a known recursion formula for ζ (2n) (see also Section 4.1): n−1

ζ (2n) =

2 X ζ (2k)ζ (2n − 2k) 2n + 1

(n ∈ N \ {1}),

(20)

k=1

which can also be used to evaluate ζ (2n) (n ∈ N \ {1}). We get no information about ζ (2n + 1) (n ∈ N) from Riemann’s functional equation, since both members of (11) vanish upon setting s = 2n + 1 (n ∈ N). In fact, until now, no simple formula analogous to (18) is known for ζ (2n + 1) or even for any special case, such as ζ (3). It is not even known whether ζ (2n + 1) is rational or irrational, except that the irrationality of ζ (3) was proven recently by Ape´ ry [56]. Instead, a known integral formula for ζ (2n + 1) is recalled here: (−1)n+1 (2π)2n+1 ζ (2n + 1) = 2(2n + 1)!

Z1

B2n+1 (t) cot(π t) dt

(n ∈ N).

(21)

0

It is readily seen that ζ (s) 6= 0 (<(s) = σ = 1), and (12) shows that ζ (s) 6= 0 (σ 5 0), except for the trivial zeros in (13). Furthermore, in view of the second series definition of ζ (s) in (1), we find that ζ (s) < 0 (s ∈ R; 0 < s < 1). The assertion that all the non-trivial zeros of ζ (s) have real part 12 is popularly known as the Riemann hypothesis, which was conjectured (but not proven) in the memoir of Riemann [977]. This hypothesis is still one of the most challenging mathematical problems today (see Edwards [398]). It is easy to derive from (12) and (13) that (cf., e.g., Srivastava [1084, p. 387, Eq. (1.15)]) ζ 0 (−2n) = lim

→0

ζ (−2n + ) (2n)! = (−1)n ζ (2n + 1)  2(2π)2n

(n ∈ N).

(22)

Relationship between ζ (s) and the Mathematical Constants B and C Just as log A in 2.1(31), we can also express log B and log C introduced in Section 1.4 as special cases of ζ 0 (s). Indeed, using the Euler-Maclaurin summation formula 1.4(68), we can obtain a number of analytical representations of ζ (s), such as

168

Zeta and q-Zeta Functions and Associated Series and Integrals

(cf. Hardy [537, p. 333]) ( n X

ζ (s) = lim

n→∞

ζ (s) = lim

k

n1−s 1 −s − − n 1−s 2

k

−s

n1−s 1 −s 1 − − n + sn−s−1 1−s 2 12

k=1

( n X

n→∞

)

−s

k=1

(<(s) > −1),

(23)

) (<(s) > −3),

(24)

and ( n X

1 n1−s 1 −s − n + sn−s−1 n→∞ 1−s 2 12 k=1  1 − (<(s) > −5). s(s + 1)(s + 2)n−s−3 720

ζ (s) = lim

k−s −

(25)

Now, it is not difficult to express the mathematical constants B and C as follows: log B = −ζ 0 (−2)

(26)

and log C = −ζ 0 (−3) −

11 , 720

(27)

respectively. P It3 is clear from 1.4(70) that log C must be the finite part of the divergent sum k log k, according to some regularization; hence, log C must be related to ζ 0 (s) for some special value of s. By differentiating both sides of (25) with respect to s and letting s = −3, we obtain −ζ 0 (−3) = lim

n→∞

( n X k=1

11 + , 720

)  3 2 4 2 4 n n 1 n n n + + − log n + − k3 log k − 4 2 4 120 16 12 

(28)

which, when compared with 1.4(70), yields the desired expression (27). The special case of (22) when n = 1 also shows that, in view of (26), log B =

ζ (3) . 4π 2

(29)

The Zeta and Related Functions

169

Integral Representations for ζ (s) In addition to (4) and (5), in terms of (2), we find from Section 2.2 that 1 ζ (s) = 0(s)

Z∞ 0

t s−1 dt et − 1

1 1 1 + ζ (s) = + 2 s − 1 0(s)

(<(s) > 1); Z∞  0

1 1 1 − + et − 1 t 2

(<(s) > −1);  π  Z∞ cosh(2π) − e−2π t dt ζ (s) = sin s 2 cosh(2πt) − cosh(2π) ts

(30) 

e−t t s−1 dt

(<(s) < 0);

(31)

(32)

0

n

ζ (s) = 1 +

X 0(k + s − 1) Bk 1 + s−1 0(s) k! k=1

1 + 0(s)

Z∞ 0

! n X 1 Bk k−1 −t s−1 − t e t dt et − 1 k!

(33)

k=0

(<(s) > −(2n − 1); n ∈ N0 ); Z∞ sin (s arctan t) dt 1 1 ζ (s) = + +2 .
1 2π t − 1 2 s−1 2 2s e 1 + t 0

(34)

Now, we recall some summation formulas analogous to the infinite series version of Plana’s summation formula 2.2(9). To do this, let z = τ + it, 1 e−2π iz − 1

= X(τ, t) + iY(τ, t)

and 1 e−π iz − eπ iz

= X1 (τ, t) + iY1 (τ, t),

from which X(τ, t) =

e2π t

cos(2πτ ) − e−2π t , − 2 cos(2πτ ) + e−2π t

sin(2πτ ) , − 2 cos(2πτ ) + e−2π t
cos(πτ ) eπt − e−π t X1 (τ, t) = 2π t , e − 2 cos(2πτ ) + e−2π t Y(τ, t) =

e2π t

170

Zeta and q-Zeta Functions and Associated Series and Integrals

and
sin(πτ ) eπ t + e−π t . Y1 (τ, t) = 2π t e − 2 cos(2πτ ) + e−2π t Conversely, we let 1 p(τ, t) := [ f (τ + it) + f (τ − it)] 2 and q(τ, t) :=

1 [ f (τ + it) − f (τ − it)], 2i

so that, by appealing to the Reflection Principle, p(τ, t) and q(τ, t) represent, respectively, the real and imaginary parts of the analytic function f (τ + it), whose domain is symmetric with respect to the τ -axis, in case the function f (τ + i0) is real. Then, we have the summation formulas (see Lindelo¨ f [769, Chapter III]): ∞ X

f (n) =

n=m

Z∞

f (τ ) dτ − 2

α

Z∞ (35)

Q(α, t) dt, 0

where, for convenience, Q(τ, t) := p(τ, t) Y(τ, t) + q(τ, t) X(τ, t); ∞ X

f (n) = −

n=m

1 2πi

α+i∞ Z  α−i∞

π sin(πz)

(36)

2 (37)

F(z) dz,

F being a primitive function of f ; ∞ X

(−1) f (n) = − n

n=m

α+i∞ Z

α−i∞

f (z) dz = −2 πiz e − e−π iz

Z∞

Q1 (α, t) dt,

(38)

0

where, for convenience, Q1 (τ, t) := p(τ, t) Y1 (τ, t) + q(τ, t) X1 (τ, t), α being any number between m − 1 and m in both (37) and (39).

(39)

The Zeta and Related Functions

171

Moreover, the formulas (35), (37) and (38) are valid, if the function f satisfies each of the following conditions: (a) f (z) is analytic for <(z) > 0 (z = τ + it); (b) limt→∞ exp(−2π |t|) f (τ + it) = 0 uniformly for 0 ≤ τ < ∞; (c) Z∞ lim

τ →∞ −∞

exp(−2π |t|) |f (τ + it)| dt = 0.

Now, we set f (z) = z−s , with m = 1 and α = 12 , in the three summation formulas just introduced. We, thus, obtain   t 2 2 − 21 s p(τ, t) = (τ + t ) , cos s arctan τ   (40) t 2 2 − 12 s sin s arctan . q(τ, t) = −(τ + t ) τ We also find, from (35), that 2s−1 ζ (s) = − 2s s−1 Setting z =

1 2

Z∞  − 1 s 2 1 + t2 sin(s arctan t) 0

dt . eπ t + 1

(41)

+ it in (37), we have

π 2s−2 ζ (s) = s−1

Z∞   1−s cos[(s − 1) arctan t] 2 1 + t2 h  i2 dt. 1 cosh π t 0 2

(42)

Starting with the second part of the definition (1) of ζ (s) and applying (38), we also get 2s−1 ζ (s) = 1 − 21−s

Z∞ − 1 s cos(s arctan t) 2  dt.  1 + t2 1 π t cosh 2 0

(43)

Formulas (34), (41), (42) and (43) are due to Jensen. It is observed that the definite integrals appearing in these four expressions represent analytic functions for all bounded values of s. In fact, any one of these expressions gives the analytic continuation of ζ (s) in the whole complex s-plane, except for a simple pole at s = 1 with its residue 1, just as it is with the contour integral representation (5). Some other integral representations of ζ (s) are recalled here as follows: 1 sin(πs) ζ (s) = + s−1 π

Z∞ dt [log (1 + t) − ψ(1 + t)] s , t 0

(44)

172

Zeta and q-Zeta Functions and Associated Series and Integrals

sin(πs) ζ (s + 1) = πs

Z∞

ψ 0 (1 + t)]

dt ts

0

(45)

Z∞ sin(πs) dt = [ψ(1 + t) + γ ] 1+s , πs t 0

and ζ (s + m) = (−1)

m−1

0(s) sin(πs) π 0(s + m)

Z∞

ψ (m) (1 + t)

dt ts

(m ∈ N),

(46)

0

all three of which are due to de Bruijn [371] and are valid for 0 < <(s) < 1. Moreover, 1 sin(πs) ζ (s) = + s − 1 π(s − 1)

Z∞ 

ψ 0 (1 + t) −

0

1 1+t



dt t1−s

(47)

(0 < <(s) < 2; s 6= 1); ζ (s) =

1 0(s + 1)

0

et ts (et − 1)2

∞
1−s −1 Z

1−2 0(s)

ζ (s) =

0
−1 Z∞ −s 1−2

ζ (s) =

2 0(s)

2 0(s + 1) π

0

t s−1 dt sinh t ts et (et + 1)2

1 2s

s(s − 1) 0

dt

1



1 2s

(<(s) > 1);

t s−1 dt et + 1

0
−1 Z∞ 1−s 1−2

ζ (s) =

ζ (s) =

Z∞

+

π 2s   0 12 s

(48)

(<(s) > 0);

(49)

(<(s) > 1);

dt

(50)

(<(s) > 0);

Z∞ 

t 2 (1−s) + t 2 s 1

1

(51)  ω(t) t

dt,

(52)

1

which is due to Riemann and where ω(t) :=

∞ X

  exp −n2 π t .

(53)

n=1

A Summation Identity for ζ (n) Sitaramachandrarao and Sivaramsarma [1035] proved the following summation identity for ζ (n), by using a known transformation formula and some reciprocity

The Zeta and Related Functions

173

relations:   q−1 n−2 ∞ X X 1 X 1  2 = n ζ (n + 1) − ζ (n − k) ζ (k + 1) qn k q=1

k=1

(n ∈ N \ {1}),

(54)

k=1

where an empty sum is understood (as usual) to be nil. The formula (54) contains many (known or new) special cases. Here, we give an elementary proof of (54) (cf. Williams [1229]). Let us, first, consider the sum: n−2 X

ζ (n − k)ζ (k + 1) = ζ (2)ζ (n − 1) + ζ (3)ζ (n − 2) + · · · + ζ (n − 1)ζ (2)

k=1

=

 ∞ X ∞  X 1 1 1 1 1 1 · + · + · · · + · p2 qn−1 p3 qn−2 pn−1 q2 p=1 q=1

= lim SN , N→∞

where, for convenience,  N X N  X 1 1 1 1 1 1 · + · + · · · + n−1 · 2 SN := p2 qn−1 p3 qn−2 p q

(N ∈ N).

(55)

p=1 q=1

Observing that the sum within braces in (55) is a geometric series and taking care of the exceptional case when p = q, the series (55) becomes   N N  X X 

q=1   p=1 p6=q

1 p−q



1−n 

q1−n p − p q

   1  + (n − 2) n+1 .  q  

(56)

Now, consider the first double sum in (56): N X N X q=1 p=1 p6=q



1 p−q

q1−n p1−n − p q



(57) =

N X N X q=1 p=1 p6=q

q1−n

1 · + p−q p

N X N X p=1 q=1 q6=p

p1−n

1 · . q−p q

174

Zeta and q-Zeta Functions and Associated Series and Integrals

Inverting the order of summation in the second term on the right-hand side of (57), it is fairly straightforward to write (57) in the form: N X N N N X N N X X X p1−n 1 X q1−n 1 + =2 . n−1 p(p − q) q(q − p) p(p − q) q p=1 q=1 q6=p

q=1 p=1 p6=q

q=1

(58)

p=1 p6=q

Combining (55) through (58), we find that  N X N  X 1 1 1 1 1 1 · + · + · · · + n−1 · 2 p2 qn−1 p3 qn−2 p q q=1 p=1

= (n − 2)

(59)

N N N X X 1 1 1 X + 2 . p(p − q) qn+1 qn−1 q=1

p=1 p6=q

q=1

Now, consider −q

N X p=1 p6=q

 N  X 1 1 1 = − p(p − q) p p−q p=1 p6=q

=

q−1 X p=1

=

q−1 X p=1

N N X X 1 1 1 1 − + − q−p p−q p q p=q+1

1 − p

1 =− + q

N−q X p=1

q−1 X p=1

1 + p

1 + p

p=1

N X p=1

(60)

1 1 − p q

N X p=N−q+1

1 , p

which, when substituted into (59), yields  N X N  X 1 1 1 1 1 1 · + · + · · · + n−1 · 2 p2 qn−1 p3 qn−2 p q q=1 p=1    q−1 N N N X X X 1 1 X 1  1  =n −2 −2 qn k qn qn+1 q=1

q=1

k=1

q=1

N X

k=N−q+1

 1 . k

Finally, we consider the inequality: 05

N X k=N−q+1

1 1 1 1 q = + + ··· + 5 , k N −q+1 N −q+2 N N −q+1

(61)

The Zeta and Related Functions

175

so that  N X 1  0< qn q=1

5

N X q=1

=

N X

k=N−q+1

1 q(N − q + 1)



N

1 1 X 1 · < k qn−1 N − q + 1 q=1

(n ∈ N \ {1}) (62)

 N  N 1 X 1 1 2 X1 + = N +1 q N −q+1 N +1 q q=1

<

q=1

2 (1 + log N) → 0 as N → ∞. N +1

By taking the limit in (61) as N → ∞ and applying (62), we complete the proof of the summation identity (54).

2.4 Polylogarithm Functions The so-called Polylogarithm functions have been studied rather extensively in the works of (for example) Lewin [751, 752], who was exceedingly fascinated by the following integral formula, even during his school days: 1 2(

√

Z 5−1)

!)2 √ 5−1 π2 − . 2 10

( log(1 − t) dt = log t

(1)

0

Many of the earlier authors who studied these functions include Euler, Abel, Legendre, Kummer, Spence and so on. These functions have been found useful in physics, electronics, quantum electrodynamics, etc., and are closely related to many other mathematical functions, such as the Clausen integral (or function) Cl2 (x) given by

Cl2 (x) = x log π − x log sin

x 2

+ 2π log

x G 1 − 2π x G 1 + 2π



,

(2)

where G is the Barnes G-function studied in Section 1.4. Throughout this work, we choose to follow the notations and conventions used by Lewin [751, 752].

176

Zeta and q-Zeta Functions and Associated Series and Integrals

The Dilogarithm Function The Dilogarithm function Li2 (z) is defined by Li2 (z) :=

∞ n X z n2

(|z| ≤ 1)

n=1

Zz =−

(3) log(1 − t) dt. t

0

It follows from (3) that   log(1 + x) − log x d 1 = Li2 − , dx x x the integration of which yields   1 1 + Li2 (−x) = 2 Li2 (−1) − (log x)2 , Li2 − x 2

(4)

where the constant of integration is determined by taking x = 1. The special case of (4) when x = −1 = eiπ gives 2 Li2 (1) = 2 Li2 (−1) +

π2 . 2

(5)

It is easy to see also that 1 Li2 (−1) = − Li2 (1), 2

(6)

which, in view of (5), yields Li2 (1) =

1 1 1 π2 + 2 + 2 + ··· = = ζ (2), 2 6 1 2 3

(7)

where ζ (s) is the Riemann Zeta function (see Section 2.3). It follows from (6) and (7) that Li2 (−1) = −

π2 . 12

(8)

By taking x = y eiπ in (4), we obtain   1 π2 1 Li2 ( y) + Li2 = − (log y)2 − iπ log y ( y > 1). y 3 2

(9)

The Zeta and Related Functions

177

From (3), we also find that Li2 (z) = − log z log(1 − z) −

Zz

log t dt 1−t

0

= − log z log(1 − z) − Li2 (1 − z) + Li2 (1), from which Li2 (z) + Li2 (1 − z) =

π2 − log z log(1 − z). 6

(10)

By choosing a suitable argument and then employing differentiation and integration, we have Landen’s formulas (see Lewin [751, p. 5]): Li2 (x) + Li2



x x−1



x x−1



1 = − {log (1 − x)}2 2

(x < 1)

(11)

and Li2 (x) + Li2



=

π2 − 2iπ log x + iπ log(x − 1) 2 1 − {log (x − 1)}2 (x > 1). 2

(12)

A factorization formula for Li2 (x) is given by n−1

 
X 1 Li2 x n = Li2 ωk x n



ω := exp

k=0



2π i n

 ,

(13)

which follows from the elementary identity: 1 − xn =

n−1 Y

1 − ωk x





ω := exp



k=0

2π i n

 .

The special case of (13) when n = 2 gives us   1 Li2 x2 = Li2 (x) + Li2 (−x). 2

(14)

By eliminating Li2 (x) between (11) and (14), we get  Li2

   x 1 1 + Li2 x2 − Li2 (−x) = − {log (1 − x)}2 . x−1 2 2

(15)

178

Zeta and q-Zeta Functions and Associated Series and Integrals

Setting x =

√ 3− 5 2

√

and x =

5−1 2

in (11) yields the following results:

Li2

( √ ! π2 1 1− 5 =− log + 2 15 2

Li2

( √ ! 1+ 5 π2 1 − =− + log 2 10 2

!)2 √ 5−1 2

(16)

and √

5+1 2

!)2 ,

(17)

respectively. The Legendre’s Chi function χ2 (x) is defined by χ2 (x) :=

∞ X n=1

x2n−1 (2n − 1)2

(−1 5 x 5 1) (18)

1 = {Li2 (x) − Li2 (−x)} , 2 which, in view of (3), immediately becomes 1 χ2 (x) = 2

Zx 0



1+t log 1−t



dt . t

(19)

From (19), one readily finds the Euler-Legendre-Landen identity: χ2



   1+x 1−x π2 1 + log x log + χ2 (x) = . 1+x 8 2 1−x

Putting (1 − x)/(1 + x) = x gives x2 + 2x − 1 = 0 or x = thus, find from (20) that

(20)   √ 2 − 1 = tan 81 π . We,

 π  π2 1 h  π i2 χ2 tan = − log tan . 8 16 4 8

(21)

Upon integrating by parts twice, it is easily observed that Zx 0

x α+1 x α+1 − 1 1 t Li2 (t) dt = Li2 (x) + log(1 − x) − α+1 (α + 1)2 (α + 1)2 α

Zx 0

1 − t α+1 dt. 1−t (22)

The Zeta and Related Functions

179

In view of 1.2(3), the integral formula (22) with x = 1 assumes the form: Z1

tα−1 Li2 (t) dt =

0

π 2 ψ(α + 1) + γ , − 6α α2

(23)

which, for α = n ∈ N0 , yields Z1

tn Li2 (t) dt =

n+1 X π2 1 1 − . 2 6(n + 1) (n + 1) k

(24)

k=1

0

Nielsen (see Lewin [751, pp. 20–21]) applied (24) to show that Z1

π2 f (tx) Li2 (t) dt = 6x

0

Zx 0

∞ 1 X an x n+1 f (t) dt − x (n + 1)2 n=0

! n+1 X 1 , k

(25)

k=1

P n −2 where f (x) := ∞ n=0 an x . Take an = n + 1, so that f (t) = (1 − t) . By making use of the well-known expansion: ∞

X xn+1 1 {log (1 − x)}2 = 2 n+1 n=1

! n X 1 , k k=1

we find the following integral equation involving the Dilogarithm function: Z1 0

π2 Li2 (t) Li2 (x) {log(1 − x)}2 dt = − − 6(1 − x) x 2x (1 − xt)2

whose special cases when x = −1 and x = Z1 0

1 2

(−1 5 x 5 1),

(26)

become

1 Li2 (t) dt = (log 2)2 2 2 (1 + t)

(27)

Li2 (t) π2 dt = , 24 (2 − t)2

(28)

and Z1 0

respectively.

180

Zeta and q-Zeta Functions and Associated Series and Integrals

If z is a pure imaginary number in (3), say z = iy ( y ∈ R), then Li2 (iy) =

  1 Li2 −y2 + iTi2 ( y) 4

( y ∈ R),

(29)

where the new function Ti2 (x) is defined by ∞ X (−1)n+1 x2n−1 (2n − 1)2

Ti2 (x) :=

(x ≤ 1).

(30)

n=1

From (29), we also see that 1 [Li2 (iy) − Li2 (−iy)] . 2i

Ti2 ( y) =

(31)

It follows from (18) that Ti2 ( y) = −i χ2 (iy).

(32)

In view of the following series expansion of tan−1 x: tan−1 x =

∞ X

(−1)n+1

n=1

x2n−1 2n − 1

(x ≤ 1),

(33)

the definition (30) can be written in the form: Ti2 (x) =

Zx

tan−1 t dt t

or

0

Zx   d(log t) Ti2 (x) = dt. tan−1 t dt

(34)

0

If one takes (34) as a definition of Ti2 (x), the domain of Ti2 (x) may be extended to ∞. Some known formulas and special values for Ti2 (x) are recalled here:   1 π Ti2 (x) − Ti2 = sgn(x) log |x| (x ∈ R), (35) x 2 where sgn(x) is defined by ( 1 (x > 0), sgn(x) := −1 (x < 0); Ti2 (1) = G, where G is the Catalan constant given in 1.3(16);   1 2x = Ti2 (x) + Ti2 (−x, 1) − Ti2 (x, 1), Ti2 2 1 − x2

(36) (37)

(38)

The Zeta and Related Functions

181

where Ti2 (x, a) is defined by Ti2 (x, a) :=

Zx 0

tan−1 t dt; a+t

(39)

√ ! √ !  1+ 3x 1 3x − x3 1− 3x Ti2 = Ti2 (x) + Ti2 √ − Ti2 √ 3 1 − 3x2 3+x 3−x √ √ !   3+x 1 + 3x π 1 1 + log −√ < x < √ , √ ·√ 6 1 − 3x 3−x 3 3 

which, upon setting x = tan θ, yields a trigonometric form: h π i h π i 1 Ti2 (tan 3θ ) = Ti2 (tan θ) + Ti2 tan − θ − Ti2 tan +θ 3 6 6 "
# π tan 6 + θ π
. + log 6 tan π6 − θ

(40)

(41)

Clausen’s Integral (or Function) From the series definition in (3) of the Dilogarithm function Li2 (z), we have ∞ ∞ X
X sin nθ cos nθ + i . Li2 eiθ = 2 n n2

(42)

n=1

n=1

From the integral definition in (3) of Li2 (z), we also have iθ

Li2 e




− Li2 (1) = −

exp(iθ Z )

log(1 − t) dt, t

1

which, upon putting t = eiη , yields iθ

Li2 e




− Li2 (1) = −i

Zθ


log 1 − eiη dη

0

Zθ = −i

     1 1 log 2 sin η exp − (π − η) i dη 2 2

0

Zθ = −i

   1 log 2 sin η dη 2

0

1 1 + (π − θ)2 − π 2 4 4

(0 5 θ 5 2π).

(43)

182

Zeta and q-Zeta Functions and Associated Series and Integrals

Comparing the real parts in (42) and (43) gives us a well-known Fourier series: ∞ X π 2 θ(2π − θ) cos nθ = − 2 6 4 n

(0 5 θ 5 2π).

(44)

n=1

Conversely, if we compare the imaginary parts in (42) and (43), we obtain Clausen’s integral (or function) Cl2 (θ) defined by [cf. Equation (2) above] θ

   Z ∞ X 1 sin nθ η dη. = − log 2 sin Cl2 (θ) := 2 n2 n=1

(45)

0

Equation (43) can, therefore, be written in the form:
π 2 θ(2π − θ) Li2 eiθ = − + i Cl2 (θ) 6 4

(0 5 θ 5 2π).

(46)

Some known properties and special values of the Clausen integral (or function) include the periodic properties: Cl2 (2nπ ± θ) = Cl2 (±θ) = ±Cl2 (θ),

(47)

which, for n = 1 and with θ replaced by π + θ , yields Cl2 (π + θ) = −Cl2 (π − θ).

(48)

From the series definition (45), it is obvious that Cl2 (nπ) = 0

(n ∈ Z),

(49)

which, for n = 1, gives Zπ



 1 log 2 sin θ dθ = 0 2

and

0

Zπ/2 1 log (sin θ) dθ = − π log 2. 2

(50)

0

Setting θ = 12 π in the series definition (45), and using (48), we find that  Cl2

   1 3 π = G = −Cl2 π , 2 2

(51)

The Zeta and Related Functions

183

where G is the Catalan constant given in 1.4(16); 1 Cl2 (2θ) = Cl2 (θ) − Cl2 (π − θ). 2

(52)

From (45), by employing integration by parts, we have  Zθ   1 1 1 Cl2 (θ) = −θ log sin θ + η cot η dη. 2 2 2 

(53)

0

If we substitute the known formula:  cot

 ∞ X 1 1 θ θ = 1 − 2 2 2 k π − 14 θ 2 2θ k=1

(54)

into (53) and evaluate the resulting integral, we obtain     ∞  X 2nπ + θ 1 2θ − 2nπ log Cl2 (θ) = −θ log sin θ + θ + 2 2nπ − θ n=1

(0 5 θ < 2π); 1 1 Ti2 (tan θ) = θ log(tan θ) + Cl2 (2θ) + Cl2 (π − 2θ), 2 2 whose interesting special case when θ = Cl2

π  6

 + Cl2

5π 6



1 12 π

(55)

(56)

becomes

4 = G. 3

(57)

The Trilogarithm Function The Trilogarithm function Li3 (z) is defined by Li3 (z) :=

∞ n X z n=1

Zz =

n3

(|z| 5 1)

Li2 (t) dt. t

(58)

0

An obvious special case of (58) is Li3 (1) = ζ (3), where ζ (s) is the Riemann Zeta function (see Section 2.3).

(59)

184

Zeta and q-Zeta Functions and Associated Series and Integrals

The following simple functional relationship for Li3 (x) would follow, if we divide both sides of (14) by x and then integrate each term:   1 Li3 x2 = Li3 (x) + Li3 (−x), 4

(60)

which, for x = 1, yields 3 Li3 (−1) = − Li3 (1). 4

(61)

Similarly, we find from (4) and (9) that Li3 (−x) − Li3



1 − x

 =−

π2 1 log x − (log x)3 6 6

(62)

and   π2 1 1 1 = log y − (log y)3 − iπ(log y)2 Li3 (y) − Li3 y 3 6 2

( y > 1),

(63)

respectively. Landen’s formula for Li3 (x) (see Lewin [751, p. 138]) is recalled here as follows:  Li3

 x π2 log(1 − x) + Li3 (1 − x) + Li3 (x) = Li3 (1) + x−1 6

1 1 − log x [log (1 − x)]2 + [log (1 − x)]3 2 6

(64)

(0 < x < 1),

whose domain can be extended by using (63) with y = 1 − x:  Li3

   x 1 π2 log(1 − x) + Li3 + Li3 (x) = Li3 (1) − x−1 1−x 6

1 1 − log(−x) [log (1 − x)]2 + [log (1 − x)]3 2 3 Taking x =

1 2

(x < 0).

in (64) or x = −1 in (65), we obtain

  1 π2 1 2 Li3 + Li3 (−1) = Li3 (1) − log 2 + (log 2)3 , 2 6 3

(65)

The Zeta and Related Functions

185

which, upon eliminating Li3 (−1) with the aid of (61), yields   7 π2 1 1 Li3 = Li3 (1) − log 2 + (log 2)3 . 2 8 12 6

(66)

Setting 1 − x = x/(1 − x), we have (1 − x)2 = x

or x2 − 3x + 1 = 0,

which gives the value: √ π  3− 5 x= = 4 sin2 2 10

(x < 1).

(67)

Now, writing 1 − x for x in (60), and evaluating (64) with x given by (67), we obtain

Li3

" √ ! √ !#3 √ ! 4 3− 5 1 3− 5 3− 5 π2 = Li3 (1) + log − log . 2 5 15 2 12 2 (68)

An interesting definite integral of Li3 (1) was given by Hjortnaes [559]: √

log( 1+2 5 )

Z

Li3 (1) = 10

t2 coth t dt,

(69)

0

which, upon writing coth t in exponential form and putting y = 1 − e−2t , yields √

5 Li3 (1) = 4

5−1

Z2

[log (1 − y)]2

0



2 1 + y 1−y

 dy.

(70)

Integrating (70) by parts and expressing it in terms of trilogarithms, we arrive, once again, at (68).

The Polylogarithm Functions The Polylogarithm function Lin (z) is defined by Lin (z) :=

∞ k X z k=1 Zz

kn

(|z| 5 1; n ∈ N \ {1})

Lin−1 (t) dt t

= 0

(71) (n ∈ N \ {1, 2}).

186

Zeta and q-Zeta Functions and Associated Series and Integrals

Clearly, we have Lin (1) = ζ (n)

(n ∈ N \ {1}),

(72)

in terms of the Riemann Zeta function ζ (s) (see Section 2.3). Setting z = iy ( y ∈ R) in (71), we have Lin (iy) =

  1 Lin −y2 + i Tin ( y) n 2

( y ∈ R),

(73)

where Tin (x) is the inverse tangent integral of order n defined by ∞ X

Tin (x) :=

(−1)k+1

k=1

x2k−1 (2k − 1)n

(|x| 5 1).

(74)

It is obvious that Tin (x) =

Zx

Tin−1 (t) dt, t

(75)

0

which, in conjunction with (34) for Ti2 (x), extends the domain of Tin (x). If z = reiθ , one defines
Lin reiθ := Lin (r, θ ) + i [Tin (%) − Tin (%, tan θ)],

(76)

where % = r sin θ/(1 − r cos θ). Then, it is readily seen that Lin (r, θ ) =

Zr

Lin−1 (t, θ ) dt t

(77)

0

and Tin (%, tan θ) =

Z%  0

Tin−1 (t) Tin−1 (t, tan θ) Tin−1 (t, tan θ) + − t + tan θ t t + tan θ

 dt.

(78)

The generalized Clausen function Cln (θ) is defined by

Cln (θ) :=

 ∞ X sin kθ       kn

(n is even),

∞  X  cos kθ     kn

(n is odd).

k=1

k=1

(79)

The Zeta and Related Functions

187

From these definitions, we have Cl2n+1 (θ) = Li2n+1 (1) −

Zθ

Cl2n (t) dt

(80)

0

and Cl2n (θ) =

Zθ

Cl2n−1 (t) dt.

(81)

0

The Log-Sine integral Lsn (θ) of order n is defined by Lsn (θ) := −

Zθ 

 1 n−1 log 2 sin t dt 2

(n ∈ N \ {1}),

(82)

0

whose special case only when n = 2 satisfies the relationship (cf. Equation (45)): Ls2 (θ) = Cl2 (θ).

(83)

The generalized Log-Sine integral of order n and index m is defined by Ls(m) n (θ) := −

Zθ t

m



 1 n−m−1 log 2 sin t dt. 2

(84)

0

The associated Clausen function Gln (θ) of order n is defined by  ∞ X cos kθ    (n is even),    kn k=1 Gln (θ) := ∞  X  sin kθ   (n is odd),   kn

(85)

k=1

which also satisfies the following obvious relationships analogous to (80) and (81): Gl2n (θ) = Li2n (1) −

Zθ

Gl2n−1 (t) dt

(86)

0

and Gl2n+1 (θ) =

Zθ

Gl2n (t) dt,

0

each of which was studied extensively by Glaisher [488].

(87)

188

Zeta and q-Zeta Functions and Associated Series and Integrals

Dividing both sides of (4) by x and integrating from 1 to x, we obtain (cf. Equation (62))   1 1 = − (log x)3 + 2 Li2 (−1) log x. (88) Li3 (−x) − Li3 − x 3! By repeating this process n − 2 times, we finally arrive at   1 1 n Lin (−x) + (−1) Lin − = − (log x)n x n! h

1

i

(89)

n

2 X (log x)n − 2k +2 Li2k (−1) (n−2k)!

(n ∈ N \ {1}),

k=1

which was attributed to Jonquie´ re by Nielsen [861], although equivalent formulas were given much earlier by Spence and Kummer (see Lewin [751, p. 173]). Taking x = eiπ in (89), we find that h



 1 + (−1)n Lin (1) = −

in π n n!

+2

1 2n

i

X in−2k π n−2k Li2k (−1). (n − 2k)!

(90)

k=1

Since the left-hand side of (90) vanishes when n is odd, the evaluation of Li2n+1 (1) is not possible from (90). Yet, using the following known relationship between Lin (1) and Lin (−1):   Lin (−1) = − 1 − 21−n Lin (1) (91) and writing 2n for n in (90), we readily get

n−1 X 2(−1)n−1 22n − 1 (−1)k−1 22k−1 − 1 1 L2n = −2 L2k , (2n)! (2n)! (2k)! (2n − 2k)!

(92)

k=1

where, for the sake of brevity, L2k := (2k)!π −2k 21−2k Li2k (1).

(93)

By using the generating function 1.6(2) of the Bernoulli numbers, one can also obtain   π 2n Li2n (−1) = (−1)n 22n−1 − 1 B2n (2n)!

(n ∈ N)

(94)

and Li2n (1) = (−1)n+1

(2π)2n B2n 2(2n)!

(n ∈ N),

which is precisely the same as 2.3(18).

(95)

The Zeta and Related Functions

189

Although there is no result like (94) when the order of the Polylogarithm function is odd, there is an analogous relation involving the Euler numbers En given in Section 1.7. Dividing (35) by x and integrating from 1 to x would yield   1 1 Ti3 (x) + Ti3 = 2Ti3 (1) + π(log x)2 x 4

(x > 0).

(96)

Repeated applications of this process lead us to Spence’s formula: Tin (x) + (−1)

n−1

  1 (log x)n−1 1 = π Tin x 2 (n − 1)! h

1 2 (n−1)

i

X (log x)n − 2k−1 Ti2k+1 (1) +2 (n − 2k − 1)!

(97) (x > 0).

k=1

Next, we find from (73) that Li2n+1 (i) − Li2n+1 (−i) = 2i Ti2n+1 (1). Setting x = exp



1 2 πi



(98)

and writing 2n + 1 for n in (89), we obtain

Li2n+1 (−i) − Li2n+1 (i) " #
n X B2k 22k−1 − 1 π 2n+1 n 2n+1 = (−1) i − + 2π . (2n + 1)! 22n+1 (2k)! (2n − 2k + 1)! 22n−2k+1 k=1

(99) Consider 1

1

1

e 2 z − e− 2 z 2e 2 z = 2 ez + 1 2



 1 2z z − , ez − 1 2 e2z − 1

(100)

which can be expanded by 1.6(2) and 1.6(40) in powers of z. Upon equating the coefficients of z2n , we find that " #
n X B2k 22k−1 − 1 E2n 1 = −2 . (101) (2n)! 22n+1 (2n + 1)! 22n+1 (2k)! (2n − 2k + 1)! 22n−2k+1 k=1

Now, a combination of (98), (99) and (101) leads us to the well-known Euler series: Ti2n+1 (1) =

∞ X k=1

(−1)k+1

1 (−1)n π 2n+1 = E2n , (2k − 1)2n+1 (2n)! 22n+2

(102)

190

Zeta and q-Zeta Functions and Associated Series and Integrals

which, in view of 1.6(59), gives Ti3 (1) =

π3 , 32

Ti5 (1) =

5π 5 , 1536

Ti7 (1) =

61 π 7 , .... 184320

(103)

Starting with the known formula for Li2 (x, θ): Li2 (r, θ) + Li2



 1 1 , θ = 2 Gl2 (θ) − (log r)2 r 2

(104)

and following a procedure analogous to that in proving (89), one obtains Lin (r, θ) + (−1) Lin n



 1 1 , θ = − (log r)n r n! h

+2

i

(105)

1 2n

X (log r)n−2k k=1

(n − 2k)!

Gl2k (θ)

(n ∈ N \{1}),

which, upon setting r = eiπ , noting that Lin (−r, θ) = Lin (r, π − θ) and writing 2n for n, gives n

Gl2n (π − θ) =

(−1)n−1 π 2n X (−1)n−k π 2n−2k + Gl2k (θ). 2(2n)! (2n − 2k)!

(106)

k=1

The following relationship between Gl2n (θ) and Gl2n (π − θ) can readily be obtained from the series definition (85): Gl2n (π − θ) + Gl2n (θ) =

1 22n−1

Gl2n (2θ).

(107)

The generating function 1.7(1) for the Bernoulli polynomials can be rewritten in the form: ∞

X 2t cos(2nπx) − 4nπ sin(2nπx) text = 1+t . t e −1 t2 + 4n2 π 2 n=1

Making use of the expansion:  2k ∞ 1 1 X t k = 2 2 (−1) 2nπ t2 + 4n2 π 2 4n π k=0

(|t| < 2nπ; n ∈ N)

(108)

The Zeta and Related Functions

191

and rearranging the double series in (108) in powers of t, the coefficients are easily expressible in terms of Gln (2πx), defined by (85). Thus, employing 1.7(1) on the lefthand side of (108) and equating the coefficients of the same powers on both sides, we find that h i 1+ 21 n n−1

Gln (2πx) = (−1)

2

πn

Bn (x) n!

(0 5 x 5 1; n ∈ N \ {1}),

(109)

which gives us the following special cases: π2 1 , Gl2 (θ) = (π − θ)2 − 4 12 1 Gl3 (θ) = θ(π − θ)(2π − θ), 12   1 2 1 4 1 2 2 Gl4 (θ) = π − θ π − πθ + θ , 90 12 4   1 Gl5 (θ) = θ(π − θ)(2π − θ) 4π 2 + 6πθ − 3θ 2 . 720

(110)

The Log-Sine Integrals A recurrence relationship for Lsn (π), defined by (82), is given by
(−1)m Lsm+2 (π) = π 1 − 2−m ζ (m + 1) m! m−1  X (−1)k+1  + 1 − 2k−m ζ (m − k + 1) Lsk+1 (π) k!

(111) (m ∈ N).

k=2

To derive the recurrence relation (111), we let Zπ I :=

   1 exp x log 2 sin θ dθ 2

0

 n ∞ Zπ n  X x 1 = log 2 sin θ dθ n! 2 n=0 0

=−

∞ n X x Lsn+1 (π). n! n=0

(112)

192

Zeta and q-Zeta Functions and Associated Series and Integrals

Alternatively, in view of 1.1(44), we have x   Zπ  1 dθ I= exp log 2 sin θ 2 0

Zπ =

2x sinx



 1 θ dθ 2

0

  1 1 0 + x √ 2 2 . = 2x π  1 0 1 + 2x By the Legendre duplication formula 1.1(29) for 0, we, thus, obtain I = π y,

(113)

where, for convenience, 0(1 + x) y := h  i2 . 1 0 1 + 2x Making use of the following notations: Dn f (x) =

dn f (x) = f (n) (x) dxn

and Dn0 f (x) =

dn f (x) = f (n) (0), n dx x=0

differentiating (112) n times and setting x = 0, we find that Lsn+1 (π) = −Dn0 I = −π Dn0 y, which, by virtue of 1.3(53), yields   Dn0 log y = (−1)n (n − 1)! 1 − 21−n ζ (n)

(114)

(n ∈ N \ {1}),

(115)

so that we have log y =: f (x) =

∞ n   X x (−1)n 1 − 21−n ζ (n), n n=2

since log y|x=0 = 0 = D10 log y.

(116)

The Zeta and Related Functions

193

From (114), we get Lsn+1 (π) = −π Dn0 exp(f (x)). Let ym := Dm e f . Then,     ym+1 = Dm+1 e f = Dm De f = Dm f 0 e f . Hence, using Leibniz’s rule for differentiation, we have

ym+1 =

m   X m

k

k=0

f (m−k+1) yk ,

(117)

which, upon considering ym |x=0 = −

1 Lsm+1 (π) π

in (114), yields (m+1) Lsm+2 (π) = −π f0 +

m   X m (m−k+1) f Lsk+1 (π). k 0

(118)

k=1

Now, setting 0

Ls2 (π) = 0,

f0 = 0

(m)

and f0

= (−1)m (m − 1)!   1 − 21−m ζ (m)

(m ∈ N \ {1})

in (118), we immediately arrive at the desired recurrence relation (111). Some simple consequences of (111) are presented below:

Ls2 (π) = −

Zπ

 1 log 2 sin θ dθ = 0, 2 

0

 2 Zπ  1 1 Ls3 (π) = − dθ = − π 3 , log 2 sin θ 2 12 0

 3 Zπ  1 3 Ls4 (π) = − log 2 sin θ dθ = π ζ (3), 2 2 0

4  Zπ  1 19 5 Ls5 (π) = − log 2 sin θ dθ = − π , 2 240 0

194

Zeta and q-Zeta Functions and Associated Series and Integrals

5  Zπ  45 5 1 Ls6 (π) = − dθ = π ζ (5) + π 3 ζ (3), log 2 sin θ 2 2 4 0

 6 Zπ  1 45 275 7 Ls7 (π) = − log 2 sin θ dθ = − π {ζ (3)}2 − π , 2 2 1344 0

7 Zπ   2835 315 3 133 5 1 dθ= π ζ (7)+ π ζ (5)+ π ζ (3), Ls8 (π) = − log 2 sin θ 2 4 8 20 0

 8 Zπ  1 log 2 sin θ Ls9 (π) = − dθ 2 0

24177 9 105 3 =− π − 1890 π ζ (3) ζ (5) − π {ζ (3)}2 . 26880 2

2.5 Hurwitz–Lerch Zeta Functions The Hurwitz–Lerch Zeta function 8(z, s, a) is defined by 8(z, s, a) :=

∞ X

zn (n + a)s

n=0 a ∈ C \ Z− ; 0 s∈C

(1)


when |z| < 1; <(s) > 1 when |z| = 1 ,

which satisfies the obvious functional relation: 8(z, s, a) = zn 8(z, s, n + a) +

n−1 X k=0

zk (k + a)s


n ∈ N; a ∈ C \ Z− 0 .

(2)

By writing the Eulerian integral 1.1(1) in the form:

0(z) = s

z

Z∞

e−st tz−1 dt

(<(z) > 0; <(s) > 0),

(3)

0

we can deduce the following integral representation from (1): 1 8(z, s, a) = 0(s)

Z∞ 0

t s−1 e−at 1 dt = 1 − ze−t 0(s)

Z∞ 0

t s−1 e−(a−1)t dt et − z

(<(a) > 0; |z| 5 1, z 6= 1, <(s) > 0; z = 1, <(s) > 1),

(4)

The Zeta and Related Functions

195

by noting that 1 zn = (n + a)s 0(s)

Z∞

e−at t s−1 ze−t


n

dt

(<(a) > 0; <(s) > 0).

(5)

0

If use is made of the infinite-series version of Plana’s summation formula 2.2(9) (cf. Lindelo¨ f [769, p. 61]; see also Erde´ lyi et al. [421, p. 22]) and the definition (1), another definite integral representation of 8(z, s, a) is obtained in the form: 1 8(z, s, a) = s + 2a

Z∞ 0

Z∞ −2 0

zt dt (t + a)s (6)

    − 1 s t dt 2 sin t log z − s tan−1 t2 + a2 2π a e t −1

(<(a) > 0),

which, for z = 1, immediately reduces to Hermite’s formula 2.2(12) for ζ (s, a). By setting z = eiθ in (4) and using (3), we get Lipschitz’s formula: 2 0(s)

∞ X n=1

einθ = (n + a)s

Z∞

e−at t s−1

0

eiθ − e−t dt cosh t − cos θ

(7)

(0 < θ < 2π; <(a) > −1; <(s) > 0). Several contour integral and series representations of 8(z, s, a) include (see Erde´ lyi et al. [421, pp. 28–31]): Z 0(1 − s) (−t)s−1 e−at 8(z, s, a) = − dt 2π i 1 − z e−t (8) C (<(a) > 0; |arg(−t)| 5 π), where the contour C is the Hankel loop of Theorem 2.5, which, obviously, does not enclose any pole of the integrand, that is, t = log z ± 2nπi (n ∈ N0 ); 8(z, s, a) =

∞ 0(1 − s) X (− log z + 2nπ i)s−1 e2nπ ia za n=−∞

(9)

(0 < a 5 1; <(s) < 0; |arg(− log z + 2nπ i)| 5 π); Lerch’s transformation formula: 8(z, s, a) =

   i(2π)s−1 0(1 − s) − 1 iπ s log z 2 L −a, 1 − s, e za 2π i     1 log z iπ s+2a −e 2 L a, 1 − s, 1 − , 2πi

(10)

196

Zeta and q-Zeta Functions and Associated Series and Integrals

where the so-called Lipschitz-Lerch transcendent L(ξ, s, a) is defined by φ(ξ, a, s) :=

∞   X e2nπ iξ 2πiξ = 8 e , s, a =: L (ξ, s, a) (n + a)s

(11)

n=0

a ∈ C \ Z− 0;

R(s) > 0

when ξ ∈ R \ Z; R(s) > 1


when ξ ∈ Z ,

which was first studied by Rudolf Lipschitz (1832–1903) and Matya´ sˇ Lerch (1860– 1922) in connection with Dirichlet’s famous theorem on primes in arithmetic progressions. Replacing s by 1 − s in 2.2(6), we obtain another equivalent form of Hurwitz’s formula 2.2(6): ζ (s, a) = 2(2π)s−1 0(1 − s)

∞ X n=1

 πs ns−1 sin 2nπa + 2

(12)

(<(s) < 0; 0 < a 5 1). By applying the binomial theorem to (− log z ± 2nπi)s−1 in (9) and making use of (12), 1.1(12) and 1.1(21), we readily obtain the following formulas (cf. Erde´ lyi et al. [421, p. 29, Eq. (8)]): 0(1 − s) 8(z, s, a) = za



1 log z

s−1 +

∞ 1 X (log z)k ζ (s − k, a) za k! k=0

(|log z| < 2π ; s 6∈ N;

(13)

a 6∈ Z− 0 );

8(z, m, a)         ∞ (log z)k (log z)m−1 1  1  X = a + ζ (m − k, a) ψ(m) − ψ(a) − log log z  k! (m − 1)! z     k=0  k6=m−1

(m ∈ N \ {1}; |log z| < 2π ; a 6∈ Z− 0 ). (14) By setting s = −m (m ∈ N) in (13) and using 2.1(17), 8 can also be expressed in terms of the Bernoulli polynomials as follows: m! 8(z, −m, a) = a z



1 log z

−m−1 −

∞ 1 X Bm+k+1 (a) (log z)k za k!(m + k + 1) k=0

(|log z| < 2π).

(15)

The Zeta and Related Functions

197

The special case of (1) when s = 1 yields 8(z, 1, a) =

∞ X zn = a−1 2 F1 (1, a ; 1 + a ; z) n+a

(|z| < 1),

(16)

n=0

and, more generally, we have 8(z, k, a) = a−k k+1 Fk (1, a, . . . , a ; a + 1, . . . , a + 1 ; z)

(k ∈ N0 ),

(17)

where p Fq denotes the generalized hypergeometric function (see Section 1.5). It follows from (13) and (16) that n o lim (1 − z)1−s 8(z, s, a) = 0(1 − s)

z→1

(<(s) < 1)

(18)

and  lim

z→1

 8(z, 1, 1) = −1. log(1 − z)

(19)

The Jonqui`ere function F(z, s) is defined by ∞ k X z F(z, s) := = z 8(z, s, 1), ks

(20)

k=1

whose many properties can be deduced from those of the 8-function. For example, Lerch’s transformation formula (10) readily yields Jonqui`ere’s formula: F(z, s) + eisπ F



   1 (2π)s 1 iπ s log z ,s = e 2 ζ 1 − s, . z 0(s) 2πi

(21)

Moreover, we obtain F(z, −m) = (−1)

m+1



1 F , −m z



(m ∈ N)

(22)

and    1 2π i log z F(z, m) + (−1) F ,m = − Bm z m! 2π i m



(m ∈ N \ {1}),

(23)

both of which provide the analytic continuation of F(z, s) in (20) outside its circle of convergence |z| = 1.

198

Zeta and q-Zeta Functions and Associated Series and Integrals

By taking a = 1 in (15) and using 1.7(5), we get   ∞ Bm+k+1 1 −m−1 X − (log z)k F(z, −m) = m! log z k!(m + k + 1) k=0

(24)

(m ∈ N; |log z| < 2π). In addition to the Jonquie` re function F(z, s) in (20), various special cases of 8(z, s, a) are listed below: ζ (s) = 8(1, s, 1); ζ (s, a) = 8(1, s, a); Li2 (z) = z8(z, 2, 1); Lis (z) :=

∞ k X z = z8(z, s, 1) = F(z, s) ks k=1

(25) (26) (27) (28)

(s ∈ C and |z| < 1; <(s) > 1 and |z| = 1).

The Taylor Series Expansion of the Lipschitz-Lerch Transcendent L(x, s, a) The function L(x, s, a), defined by (11), was first studied by Lipschitz [770, 771] and Lerch [745] in connection with Dirichlet’s famous theorem on primes in arithmetic progressions. For x ∈ Z, (11) reduces immediately to the Hurwitz Zeta function ζ (s, a) (see Section 2.2). Since L(x, s, a) is a special case of 8(z, s, a), many properties of this function can be obtained from those of 8(z, s, a). The Lerch functional equation for L(x, s, a) can be deduced from (10):      1 0(s) π i 2 s−2ax π i − 12 s+2a(1−x) L(x, 1 − s, a) = L(−a, s, x) + e L(a, s, 1 − x) e (2π)s (0 < x < 1; 0 < a < 1). (29)

Several interesting proofs of (29) were given by Apostol [58] and Berndt [117]. The function L(x, s, a) satisfies the differential-difference equations: ∂L(x, s, a) = −s L(x, s + 1, a) ∂a

(30)

∂L(x, s, a) + 2πia L(x, s, a) = 2πi L(x, s − 1, a). ∂x

(31)

and

The Zeta and Related Functions

199

Klusch [673] gave a Taylor series expansion of the function L(x, s, a + ξ ) in the neighborhood of ξ = 0 in the form: For fixed a ∈ R+ , ∞ X

(−1)k

k=0

(s)k L(x, s + k, a) ξ k = L(x, s, a + ξ ) k!

(|ξ | < a),

(32)

which provides the analytic continuation of L(x, s, a) to the whole complex s-plane. Klusch [673] also applied (32) to derive several summation formulas which will be investigated systematically in Chapter 3. By setting ξ = −t in (32), we get ∞ X (s)k L(x, s + k, a) tk = L(x, s, a − t) k!

(|t| < a),

(33)

k=0

which, upon replacing k by k + 2 and s by s − 1, dividing the resulting equation by t2 and differentiating both sides with respect to t yields the following corrected version of one of Klusch’s results (Klusch [673, p. 517, Eq. (3.3)]): ∞ X k=1

k 0(s + k + 1) L(x, s + k + 1, a) tk−1 (k + 2)! r0(s)

= t−2 {L(x, s, a − t) + L(x, s, a)} −

2t−3 {L(x, s − 1, a − t) − L(x, s − 1, a)} rs − 1 (|t| < a; t 6= 0). (34)

Evaluation of L(x, −n, a) Since L(x, s, a) = ζ (s, a) (x ∈ Z), whose properties are studied in Section 2.2, in what follows, we assume that x is not an integer. By making use of the classical method presented in Section 2.1, L(x, s, a) can be extended to the whole complex s-plane by means of the contour integral (see Apostol [58]): I(x, s, a) =

1 2π i

Z C1

zs−1 eaz dz, 1 − ez+2π ix

(35)

where the contour C1 is a loop which begins at −∞, encircles the origin once in the positive (counter-clockwise) direction and returns to −∞. Since I(x, s, a) is an entire function of s and L(x, s, a) = 0(1 − s) I(x, s, a),

(36)

we can use (36) to continue L(x, s, a) analytically to the whole complex s-plane as an entire function of s (x 6∈ Z).

200

Zeta and q-Zeta Functions and Associated Series and Integrals

Apostol [58] computed recursively the values of L(x, −n, a) (n ∈ N0 ; x 6∈ Z) by applying (31) and (35). Thus, he obtained (cf. Apostol [58]) L(x, 0, a) =

i 1 1 = cot(πx) + , 1 − exp(2πix) 2 2

a 1 L(x, −1, a) = [i cot(πx) + 1] − csc2 (πx), 2 4   2 a 1 a i L(x, −2, a) = i cot(πx) + − csc2 (πx) − cot(πx) csc2 (πx) 2 4 2 4

(37) (38) (39)

and Bn+1 a; e2πix L(x, −n, a) = − n+1


(n ∈ N0 ),

(40)

where the Bn (a; α) are defined by 1.8(1).

2.6 Generalizations of the Hurwitz–Lerch Zeta Function Lin and Srivastava [765] introduced and investigated the following generalization of the Hurwitz–Lerch Zeta function 8(z, s, a), defined by 2.5(1): ) 8(ρ,σ µ,ν (z, s, a) :=

∞ X (µ)ρ n zn (ν)σ n (n + a)s n=0

+ µ ∈ C; a, ν ∈ C \ Z− 0 ; ρ, σ ∈ R ; ρ < σ

when s, z ∈ C;

(1)

ρ = σ and s ∈ C when |z| < δ := ρ −ρ σ σ ; ρ = σ and <(s − µ + ν) > 1
when |z| = δ , where (λ)n denotes the Pochhammer symbol, defined by 1.1(5). Clearly, we have ) (0,0) 8(σ,σ ν,ν (z, s, a) = 8µ,ν (z, s, a) = 8(z, s, a)

(2)

and (1,1)

8µ,1 (z, s, a) = 8∗µ (z, s, a) :=

∞ X (µ)n zn , n! (n + a)s

(3)

n=0

where, as also noted by Lin and Srivastava [765], 8∗µ (z, s, a) is a generalization of the Hurwitz–Lerch Zeta function considered by Goyal and Laddha [504, p. 100, Eq. (1.5)].

The Zeta and Related Functions

201

For further results involving these classes of generalized Hurwitz–Lerch Zeta functions, see the recent works by Garg et al. [467] and Lin et al. [767]. Recently, among other things, Choi et al. [273] presented and investigated the following multiple-series generalization of the Hurwitz–Lerch Zeta function 8(z, s, a), by calling it the multiple (or, simply, n-tuple) Hurwitz–Lerch Zeta function 8n (z, s, a), defined by ∞ X

8n (z, s, a) :=

m1 ,..., mn =0

zm1 +···+mn (m1 + · · · + mn + a)s

(4)


a ∈ C \ Z− 0 ; s ∈ C when |z| < 1; <(s) > n when |z| = 1 . In its special case when z = 1, the definition (4) yields the following familiar multiple (or, simply, n-tuple) Hurwitz Zeta function ζn (s, a) given in 2.1(2):
8n (1, s, a) = ζn (s, a) <(s) > n; a ∈ C \ Z− 0 .

(5)

The special case of (4) when n = 1 reduces to the Hurwitz–Lerch Zeta function 8(z, s, a), defined by 2.5(1), as follows: 81 (z, s, a) = 8(z, s, a).

(6)

Furthermore, when n = 1 and z = 1, the definition (4) yields the Hurwitz (or generalized) Zeta function ζ (s, a), defined by 2.2(1): 81 (1, s, a) = ζ (s, a)


a ∈ C \ Z− 0 ; <(s) > 1 .

(7)

See also a recent investigation by Srivastava and Attiya [1092], using the Hurwitz– Lerch Zeta function in Geometric Function Theory. We recall several interesting properties of the multiple Hurwitz–Lerch Zeta function 8n (z, s, a) in (4) investigated by Choi et al. [267, 273]. First of all, the multiple Hurwitz–Lerch Zeta function 8n (z, s, a) satisfies the following functional relation: 8n (z, s, a) = z` 8n (z, s, ` + a) +

X m1 ,..., mn =0 m1 +···+mn 5`−1

zm1 +···+mn (m1 + · · · + mn + a)s (` ∈ N; a ∈ C \ Z− 0 ),

which, for n = 1, yields a known functional relation for 8(z, s, a) given by 2.5(2).

(8)

202

Zeta and q-Zeta Functions and Associated Series and Integrals

It is easy to derive the following summation formula: ∞ X (s)k k=0

k!

8n (z, s + k, a)tk = 8n (z, s, a − t)

(|t| < |a|; s 6= 1),

(9)

where (λ)m denotes the Pochhammer symbol defined by 1.1(5). Indeed, we see that 8n (z, s, a − t) =

zm1 +···+mn (m1 + · · · + mn + a − t)s

X m1 ,...,mn =0

=

 −s zm1 +···+mn t 1− (m1 + · · · + mn + a)s m1 + · · · + mn + a

X m1 ,...,mn =0

=

∞

X (s)k zm1 +···+mn s (m1 + · · · + mn + a) k!

X m1 ,...,mn =0

k=0



t m1 + · · · + mn + a

k

(|t| < |m1 + · · · + mn + a|; mj ∈ N0 ; j = 1, . . . , n) =

∞ X (s)k k=0

=

k!

∞ X (s)k k=0

k!

 X  m1 ,...,mn =0

 zm1 +···+mn  tk (m1 + · · · + mn + a)s+k

8n (z, s + k, a)tk

(|t| < |a|; s 6= 1),

which proves (9). The interchange of the order of summation can be justified by the absolute convergence of the series involved. Now, from the Eulerian integral form of the Gamma function 0 (see Section 1.1), we have

0(s) = u

s

Z∞

e−ut t s−1 dt


<(s) > 0; <(u) > 0 .

(10)

0

By setting u = m1 + · · · + mn + a


(m1 , . . . , mn ) ∈ N0 n ,

multiplying each side of the resulting equation by zm1 +···+mn and summing up for each mj from 0 to ∞, ( j = 1, . . . , n), we can get the following integral representation for

The Zeta and Related Functions

203

8n (z, s, a): 1 8n (z, s, a) = 0(s)

Z∞ 0

t s−1 e−at 1
n dt = 0(s) 1 − z e−t

Z∞ 0

t s−1 e−(a−n)t dt (et − z)n

(11)


n ∈ N; <(a) > 0; |z| 5 1 (z 6= 1) and <(s) > 0; z = 1 and <(s) > 1 . In its special case when n = 1, (11) yields a known integral representation for the Hurwitz–Lerch Zeta function 8(z, s, a) (see 2.5(4)). We can get a particular infinite-series version of Plana’s summation formula 2.2(9) in the form: ∞ X k=0

1 f (k) = f (0) + 2

Z∞

f (τ ) dτ − 2

0

Z∞ 0

q(0, t) dt, e2π t − 1

(12)

where f (z) is a bounded analytic function in 0 5 <(z) < ∞, the integral Z∞

f (τ ) dτ

0

is assumed to be convergent, and q(n, t) → 0

as

n→∞

or

t → ∞.

If we apply (12) to the function zu+m1 +···+mn−1 , (u + m1 + · · · + mn + a)s

f (u) := then we get ∞ X k=0

zk+m1 +···+mn−1 zm1 +···+mn−1 = s (k + m1 + · · · + mn−1 + a) 2 (m1 + · · · + mn−1 + a)s Z∞

+ 0

zτ +m1 +···+mn−1 dτ (τ + m1 + · · · + mn−1 + a)s

Z∞ −2

h i− s 2 zm1 +···+mn−1 (m1 + · · · + mn−1 + a)2 + t2

0

  · sin t log z − s arctan

t m1 + · · · + mn−1 + a



dt . −1

e2π t

204

Zeta and q-Zeta Functions and Associated Series and Integrals

Now, summing up for each mj from 0 to ∞, ( j = 1, . . . , n − 1), we finally obtain the following integral representation for 8n (z, s, a): 1 8n (z, s, a) = 8n−1 (z, s, a) + 2

Z∞

zt 8n−1 (z, s, a + t) dt

0 ∞ X

−2

zm1 +···+mn−1

m1 ,...,mn−1 =0

Z∞ h i− s 2 (m1 + · · · + mn−1 + a)2 + t2

(13)

0

  · sin t log z − s arctan

t m1 + · · · + mn−1 + a



dt −1

e2π t

(n ∈ N),

provided that the involved integrals and series converge. The special case of (13) when n = 1 yields the known integral representation for 8(z, s, a) (see 2.5(6)). It is not difficult to deduce the following contour integral representation of 8n (z, s, a), by applying a procedure that is used to get a contour integral representation of the multiple Hurwitz Zeta function ζn (s, a) (see, for details, Section 2.1): 0(1 − s) 8n (z, s, a) = − 2π i

(−t)s−1 e−at
n dt 1 − z e−t C
n ∈ N; <(a) > 0; |arg(−t)| 5 π , Z

(14)

where the contour C is the Hankel loop given as in 2.1(7). The special cases of (14) when z = 1 and n = 1 would immediately yield the known integral representations of the multiple Hurwitz Zeta function ζn (s, a) in 2.1(8) and the Hurwitz–Lerch Zeta function 8(z, s, a) in 2.5(8), respectively. The special case of 8n (z, s, a) in (14) when s = −`

(` ∈ N0 ) (α)

can be expressed in terms of the Apostol-Bernoulli polynomials Bk (a; λ) of order α, defined by means of the following generating functions (see Section 1.8): 8n (z, −`, a) = (−1)n Indeed, upon setting s = −`

(` ∈ N0 )

`! (n) B (a; z) (n + `)! n+`

(` ∈ N0 ).

(15)

The Zeta and Related Functions

205

in (14), we find, by using 1.8(13), that 8n (z, −`, a) = −

0(1 + `) 2πi

(−t)−`−1 e−at
n dt 1 − z e−t

Z C

= (−1)` `! Res t−n−`−1



t=0

`

= (−1) `! Res t

−n−`−1

t=0

= (−1)n

t 1 − z e−t

∞ X

(n)

n

Bk (a; z)

k=0

e−at (−1)k tk k!

`! (n) B (a; z). (n + `)! n+`

The special case of (15) when z = 1, together with 1.8(14), would reduce immediately to the known formula (see 2.1(16)). The multiple Hurwitz–Lerch Zeta function 8n (z, s, a) can, thus, be expressed as follows, as a simple series (see 2.1(19)): 8n (z, s, a) =

 ∞  X m+n−1 n−1

m=0

zm (m + a)s

(16)


a ∈ C \ Z− 0 ; s ∈ C when |z| < 1; <(s) > n when |z| = 1 . We find from (16), as in getting 2.1(21), that 8n (z, s, a) is expressible as a finite combination of the Hurwitz–Lerch Zeta function 8(z, s, a) with polynomial coefficients in a as follows: 8n (z, s, a) =

n−1 X

pn,j (a) 8(z, s − j, a),

(17)

j=0

where   n−1 X 1 n+1−j ` pn,j (a) = (−1) s(n, ` + 1) a`−j . (n − 1)! j

(18)

`=j

We now find pn,j (a) in (18) as a polynomial in a of degree n − 1 − j with rational coefficients. Since ∞ X `=0

f (`) =

k−1 X ∞ X j=0 `=0

f (k` + j)

(k ∈ N),

(19)

206

Zeta and q-Zeta Functions and Associated Series and Integrals

we get

8n (z, s, a) = k−s

k−1 X j=0

  a+j j 8n zk , s, z k

(k ∈ N).

(20)

Thus, by combining (11) and (20), we immediately obtain the following sum-integral representation: Z∞ s−1 −(a+j)t k−1 X t e zj
dt 8n (z, s, a) = k e−kt n 0(s) 1 − z j=0

(21)

0


k, n ∈ N; <(a) > 0; <(s) > 0 when |z| 5 1 (z 6= 1); <(s) >1 when z = 1 , which is a straightforward generalization of the following sum-integral representation given by Lin and Srivastava [765, p. 727, Eq. (7)]: Z∞ s−1 −(a+j)t k−1 X t e zj
dt 8(z, s, a) = 0(s) 1 − zk e−kt j=0

(22)

0


k ∈ N; <(a) > 0; <(s) > 0 when |z| 5 1 (z 6= 1); <(s) > 1 when z = 1 . For further special cases of (21) or (22), just as noted also by Lin and Srivastava [765, p. 726], see Yen et al. [1244, p. 100, Theorem] and Nishimoto et al. [864, p. 94, Theorem 4]. In view of (17), we can deduce some properties of the multiple Hurwitz–Lerch Zeta function 8n (z, s, a) from those (see, e.g., [447]) of the Hurwitz–Lerch Zeta function 8(z, s, a). Combining the integral formula 2.5(6) for 8(z, s, a) with (17), we obtain another sum-integral representation for 8n (z, s, a): 8n (z, s, a) Z∞ n−1 n−1   n−1 X X ` (−1)n+1 X zt j ` (−1) s(n, ` + 1) a + pn,j (a) = s dt 2 a (n − 1)! j (t + a)s−j j=0

−2

n−1 X j=0

pn, j (a)

Z∞ 0

`=j

j=0

0

     1 ( j−s) dt 2 −1 t sin t log z + ( j − s) tan t2 + a2 2π a e t −1
n ∈ N; <(a) > 0; s ∈ C . (23)

The Zeta and Related Functions

207

By applying Apostol’s formula 2.5(40) (see [58, p. 164]; see also [467, p. 806, Eq. (19)]):
  B` a; e2π iξ 2π iξ φ(ξ, a, 1 − `) = 8 e , 1 − `, a = − `

(` ∈ N; ξ ∈ R)

(24)

to (17), we obtain
n−1   X B`+j a; e2π iξ 8n e2π iξ, 1 − `, a = − pn,j (a) `+j

(n, ` ∈ N).

(25)

j=0

The following known summation formula for 8(z, s, a), expressed in terms of the Hurwitz (or generalized) Zeta function ζ (s, a) (see [788, p. 298, Eq. (38)]):     q   X a+`−1 2(` − 1)pπi 2π i p 8 e q , s, a = q−s exp ζ s, q q

(p ∈ Z; q ∈ N),

`=1

(26) when applied to (17), would readily yield     q n−1   X X a+`−1 2(` − 1)pπi 2π i qp −s j 8n e , s, a = q exp pn,j (a) q ζ s − j, q q `=1

j=0

(n, q ∈ N; p ∈ Z). (27) Lin and Srivastava [765, p. 729, Eq. (17)] presented the following integral representation for 8∗µ (z, s, a):

8∗µ (z, s, a) =

1 0(s)

Z∞ 0

t s−1 e−at
µ dt 1 − z e−t

(28)


<(a) > 0; <(s) > 0 when |z| 5 1 (z 6= 1); <(s) > 1 when z = 1 , which was given earlier by Goyal and Laddha [504, p. 100, Eq. (1.6)], together with the seemingly unnecessary constraint µ = 1 (see, for details, [765, p. 729]). Here, in view of (11) and (28), it is seen that 8n (z, s, a) = 8∗n (z, s, a)

(n ∈ N).

(29)

208

Zeta and q-Zeta Functions and Associated Series and Integrals

Lin and Srivastava [765] also recalled the Riemann-Liouville fractional derivative µ operator Dz , defined by (cf. [422, p. 181 et seq.]; see also [896] and [648])

Dzµ { f (z)} :=

 Zz 
1   (z − t)−µ−1 f (t) dt R (µ) < 0    0 (−µ) 0

   m    d Dzµ−m { f (z)} dzm


m − 1 5 R (µ) < m (m ∈ N) , (30)

to, first, notice that 8∗µ (z, s, a) is essentially a Riemann-Liouville fractional derivative of the classical Hurwitz–Lerch function 8 (z, s, a) of order µ − 1 as follows: n o
1 8∗µ (z, s, a) = Dzµ−1 zµ−1 8 (z, s, a) R (µ) > 0 . (31) 0 (µ) By using their remarkable observation (31), together with the Eulerian integrals of the first and second kind, Hankel’s integral representation for the Gamma function [421, p. 13, Eq. 1.6(2)] and the theory of the Mellin-Barnes contour integration (cf. [421, p. 49]), Lin and Srivastava [765] presented various integral representations for 8∗µ (z, s, a). In view of (29), as well as for the sake of completeness, we recall the known integral representations for 8∗µ (z, s, a) given by Lin and Srivastava (see [765, Eqs. (27), (28), (34), and (37)]) in their following special forms for 8n (z, s, a): 1 8n (z, s, a) = (n − 1)!

Z∞

(0,1)

tn−1 e−t 8n,1 (zt, s, a) dt

(n ∈ N),

(32)

0

   Z1  1 1 n−1 (0,1) 1 8n (z, s, a) = log 8n,1 z log , s, a dt (n − 1)! t t

(n ∈ N),

(33)

0

8n (z, s, a) =

1 2πi

(+0) Z

(0,1)

w−1 ew 8n,1

  z w−1 , s, a dw (n ∈ N; | arg(w)| 5 π),

−∞

(34) where the contour of integration in the complex w-plane is Hankel’s loop (cf., e.g., [1225, p. 245]), which starts from the point at −∞ along the lower side of the negative real axis, encircles the origin once in the positive (counter-clockwise) direction and then returns to the point at −∞ along the upper side of the negative real axis. 1 8n (z, s, a) = 2πi

τZ +i∞

(1,0)

w−1 ew 8n,1

  zw−1 , s, a dw


n ∈ N; τ ∈ R+ ,

τ −i∞

where the contour of integration is a Mellin-Barnes contour (cf. [421, p. 49]).

(35)

The Zeta and Related Functions

209

Garg et al. [467, p. 809, Theorem 2; p. 811, Theorem 3] gave transformation formulas for 8∗n (z, s, a) and Lin et al. [767, p. 825, Eqs. (39) and (40)] presented several expansion and transformation formulas for the general case 8∗µ (z, s, a). By means of (29), we choose to rewrite these known results in their special cases applicable to 8n (z, s, a) as follows:  n−1  i z−a 0(1 − s) X n − 1 (n) Bn−k−1 8n (z, s, a) = (n − 1)! k k=0

k X

   k s−1 · (−1)n−k+j−1 j! (−a)k−j (2π)s−j−1 j j j=0      log z 1 · exp − (s − j)πi 8 e−2π ai , 1 − s + j, 2 2πi      1 log z − exp 2a + (s − j) π i 8 e2πai , 1 − s + j, 1 − 2 2πi

(n ∈ N), (36)

provided that each side of (36) exists;  j  n−1 X ( j−a+1)n−j−1 X j − 1 ( j) (1−s)k Bj−k (2π)s−k−1 j!(n − j − 1)! k−1 j=0 k=0      log z 1 −2π ai , 1 − s + k, · exp − (s − k)πi 8 e 2 2πi      1 log z 2π ai − exp 2a + (s − k) πi 8 e , 1 − s + k, 1 − (n ∈ N), 2 2πi (37)

8n (z, s, a) = i z−a 0(1−s)

provided that each side of (37) exists;

8n (z, s, a) = i z−a 0(1 − s)

n−1 X ( j − a + 1)n−j−1 j=0

 j  X j−1

j! (n − j − 1)!

( j)

(1 − s)k Bj−k ·(2π)s−k−1 k−1 k=0      1 log z −2π ai · exp − (s − k)πi 8 e , 1 − s + k, 2 2πi      1 log z 2π ai , 1 − s + k, 1 − − exp 2a + (s − k) πi 8 e 2 2πi ·

(n ∈ N), (38)

210

Zeta and q-Zeta Functions and Associated Series and Integrals

and     n−1  i 0(1 − n + s) X n − 1 p p = j− +1 8n e2π iξ, n − s, q (n − 1)! j q n−j−1 j=0

j  X

 j−1 ( j) (1 − n + s)k Bj−k ·(2πq)n−s−k−1 · k−1 k=0 " q       X ξ + r −1 1 2(ξ + r −1)p · ζ 1− n + s + k, exp − (n−s−k)+ πi q 2 q r=1    #  q X 1 2(r − ξ )p r−ξ exp (n − s − k) + πi − ζ 1 − n + s + k, q 2 q r=1

(p ∈ Z; q, n ∈ N). (39) Garg et al. [467, p. 813, Eqs. (51) and (52)] applied the definition 2.5(11) and the summation formula (26) to (36) and (37), by setting z = e2π iξ

and

a=

p q

(p ∈ Z; q ∈ N; ξ ∈ R),

and, thereby, expressed the function:   p 8∗n e2π iξ, n − s, q in terms of the Hurwitz Zeta function ζ (s, a) and the generalized Apostol-Bernoulli ( j) polynomials Bk (x; λ). We rewrite these known  results of Garget al. [467, Eqs. (51) and (52)] in their following special forms for 8n e2π iξ, n − s, pq ):

   n−1  p i (−1)n−1 0(1 − n + s) X n − 1 (n) 2π iξ 8n e , n − s, = Bn−k−1 qn−s−k−1 q (n − 1)! k k=0

·

 k   X k n−s−1 j=0 q X

j

j

j! pk−j (2π)n−s−j−1

"

      ξ +r−1 1 2(ξ + r − 1)p · ζ 1−n + s + j, exp − (n−s−j) + πi q 2 q r=1    #  q X 1 2(r − ξ )p r−ξ − ζ 1 − n + s + j, exp (n − s − j) + πi q 2 q r=1

(n ∈ N) (40)

The Zeta and Related Functions

211

and     n−1  i 0(1 − n + s) X n − 1 p p = j− +1 8n e2π iξ, n − s, q (n − 1)! j q n−j−1 j=0

·

 j  X j−1 k=0

k−1

( j)

(1 − n + s)k Bj−k ·(2πq)n−s−k−1

      q X ξ +r−1 1 2(ξ +r−1)p · ζ 1−n + s + k, exp − (n−s−k) + πi q 2 q "

r=1

   #  q X 1 2(r − ξ )p r−ξ exp πi (n − s − k) + − ζ 1 − n + s + k, q 2 q r=1

(n ∈ N). (41) In a recent paper, Srivastava et al. [1107] introduced and investigated further (ρ,σ ) generalizations of the above-defined Lin-Srivastava model 8µ,ν (z, s, a), first, in the following form: (ρ,σ,κ)

8λ,µ;ν (z, s, a) :=

∞ X (λ)ρn (µ)σ n n=0

zn (ν)κn · n! (n + a)s

+ λ, µ ∈ C; a, ν ∈ C \ Z− 0 ; ρ, σ, κ ∈ R ; κ − ρ − σ > −1 when s, z ∈ C;

κ − ρ − σ = −1 and s ∈ C when |z| < δ := ρ

−ρ

σ

(42)

κ

κ ;
κ − ρ − σ = −1 and <(s + ν − λ − µ) > 1 when |z| = δ ∗ ∗

−σ

and, then, in a substantially more general form given by p Q (ρ ,...,ρ ,σ ,...,σ ) 8λ11,...,λpp;µ11,...,µqq (z, s, a) =

∞ X n=0

(λj )nρj

j=1 q Q

n!

(µj )nσj

zn (n + a)s

j=1

 p, q ∈ N0 ; λj ∈ C ( j = 1, . . . , p); a, µj ∈ C \ Z0− ( j = 1, . . . , q); ρj , σk ∈ R+ ( j = 1, . . . , p; k = 1, . . . , q); 1 > −1 when s, z ∈ C;  1 1 = −1 and s∈C when |z|<∇; 1 = −1 and <(4)> when |z| = ∇ , 2 (43)

212

Zeta and q-Zeta Functions and Associated Series and Integrals

where, for convenience,    q p Y Y σj −ρj ∇ :=  ρ   σ , j

j

j=1

j=1

1 :=

q X

σj −

j=1

p X

ρj

j=1

and 4 := s +

q X

µj −

j=1

p X

λj +

j=1

p−q . 2

The special case of the definition (43) when p − 1 = q = 1 would obviously correspond to the above-investigated (Lin-Srivastava) generalized Hurwitz–Lerch Zeta (ρ,σ,κ) function 8λ,µ;ν (z, s, a), defined by (42). The following further special case of the generalized Hurwitz–Lerch Zeta function (ρ,σ,κ) 8λ,µ;ν (z, s, a) (which corresponds to the definition (43) with p − 2 = q = 1) when ρ = σ = κ = 1: 8λ,µ;ν (z, s, a) :=

∞ X (λ)n (µ)n n=0

zn (1,1,1) =: 8λ,µ;ν (z, s, a) (ν)n · n! (n + a)s

(44)

λ, µ ∈ C; ν, a ∈ C \ Z− 0 ; s ∈ C when |z| < 1; <(s + ν − λ − µ) > 1
when |z| = 1 was investigated earlier by Garg et al. [466, p. 313, Eq. (1.7)]. For various interesting properties and results involving the families of generalized Hurwitz–Lerch Zeta functions: (ρ,σ,κ)

(ρ ,...,ρ ,σ ,...,σ )

8λ,µ;ν (z, s, a) and 8λ11,...,λpp;µ11,...,µqq (z, s, a), which are defined by (42) and (43), respectively, as well as for the numerous special cases and consequences of each of these general models, we choose to refer the interested reader to the above-mentioned paper by Srivastava et al. [1107], as well as to a subsequent investigation by Srivastava et al. [1106] in which several two-sided bounding inequalities for the extended Hurwitz–Lerch Zeta function (ρ ,...,ρ ,σ ,...,σ )

8λ11,...,λpp;µ11,...,µqq (z, s, a) were given (see also Problems 51, 52 and 53 at the end of this chapter).

The Zeta and Related Functions

213

2.7 Analytic Continuations of Multiple Zeta Functions The multiple Hurwitz zeta function ζn (s, a) (n ∈ N), defined by 2.1(2), is obviously a generalization of the Riemann zeta function ζ (s) given in 2.3(1). Another very useful and widely-investigated generalization of ζ (s) is the multiple zeta functions ζd (s1 , . . . , sd ) of depth d (d ∈ N), defined by 1

X

ζd (s1 , . . . , sd ) :=

0


(1)

n1 s1 n2 s2 · · · nd sd

(s1 , . . . , sd ) ∈ C ; < (sd ) > 1; d

d X



< sj > d ,


j=1

which is often referred to as Euler-Riemann-Zagier zeta functions of depth d. If (s1 , . . . , sd ) ∈ Nd , ζd (s1 , . . . , sd ) is also called the multiple zeta values (or EulerZagier sums) (see [1246]). Since these values have turned out to be connected with a variety of research subjects, such as topology, theoretical physics and the theory of mixed Tate motives, an extensive research has been made of those values (see, e.g., [151], [182], [734], [1245], [1246], [1247]). Ohno [872] presented a unified algebraic relationship among the multiple zeta values that generalizes the so-called sum formula (or sum conjecture) (see [560]) and another remarkable identity, referred to as the duality theorem (see [1246]). Here, we introduce two totally different methods of analytic continuation of ζd (s1 , . . . , sd ), defined by (1), which were given by Zhao [1260] and Akiyama et al. [16]. An analytic continuation of ζ2 (s1 , s2 ) was proven by, for example, Apostol and Vu [68], Atkinson [78], Motohashi [849] and Katsurada and Matsumoto [639]. Arakawa and Kaneko [70] gave the analytic continuation of ζd (s1 , . . . , sd ) by considering a function of one variable sd , while s1 , . . . , sd−1 are positive integers.

Generalized Functions of Gel’fand and Shilov We remark in proceeding this subsection that most of the arguments given here are due to Zhao [1260]. Theorem 2.7 The infinite sum in (1) converges absolutely when < (sd ) > 1

and

d X


< sj > d.

j=1

Proof. The case d = 1 is obvious, since ζ (s1 ) becomes the Riemann zeta function, defined by 2.3(1), and converges absolutely for < (s1 ) > 1. So, let us assume d = 2 and write σi = <(si ). If σ1 > 1, then, being also σ2 > 1, the result is clear: |ζ2 (s1 , s2 )| 5

X 0
∞ ∞ X 1 X 1 1 5 · < ∞. n1 σ1 n2 σ2 n1 σ1 n2 σ1 n1 =1

n2 =1

214

Zeta and q-Zeta Functions and Associated Series and Integrals

If σ1 = 1, then the theorem is proven by using the following well-known asymptotic formula (see 1.2(68)): n X 1 = log n + O(1) n1

(n → ∞).

(2)

n1 =1

Since σ2 > 1, write σ2 = δ + µ for some δ > 1 and µ > 0. Then, we have, as n → ∞, n X n1 =1

n 1 X 1 1 1 = = σ (log n + O(1)) σ σ n1 ·n 2 n 2 n1 n 2 n1 =1

=

1 1 L · [log n + O(1)] 5 δ nδ nµ n

for some L > 0, the second term in the third equality tending to as n → ∞. We, therefore, find that |ζ2 (s1 , s2 )| 5 L

∞ X 1 < ∞. nδ n=1

If σ1 < 1, then one has the following estimate: n−1 X n1 =1

[log 2 n] X 1 = n1 σ1 ·nσ2 k=1

n 2k

5n1 <

< 2max{0, −σ1 } =

1

X n 2k−1

∞ X

k=1 ∞ } max{0, −σ X 1 2

nσ1 +σ2 −1

k=1

n1

σ1 nσ2

n/2k
σ n/2k 1 nσ2 1 M 5 σ +σ −1 n 1 2 2(1−σ1 )k

for some M > 0, being 1 − σ1 > 0. So, in this case, being σ1 + σ2 − 1 > 1, |ζ2 (s1 , s2 )| 5 M

∞ X n=1

1 nσ1 +σ2 −1

< ∞.

The general case is seen to be easily verified by following the argument of case d = 2. 

Preliminary on Generalized Functions For any n ∈ N, Zhao [1260] introduces the space K of test functions consisting of all the complex-valued smooth functions on Rn , which decrease to zero faster than any

The Zeta and Related Functions

negative power of v u n uX xi 2 |x| = t

215

as |x| → ∞

x = (x1 , . . . , xn ) ∈ Rn




i=1

(for details of test functions, see [478, Chapter 1]). Throughout this subsection, we use the abbreviations x = (x1 , . . . , xn ) ∈ Rn and s = (s1 , . . . , sn ) ∈ Cn . We say that a generalized function g(x; s) (for this definition, see [478, Chapter 1]) is entire in s, if the inner product < g(x; s), ϕ(x) > is an entire function of s for any test function ϕ(x). As an example of the generalized functions, let us suppose, for x ∈ R, that  s  x (x > 0) s (3) x+ =  0 (x 5 0), whose value on a test function ϕ(x) is given by s < x+ , ϕ(x) >:=

Z∞

xs ϕ(x) dx.

0

The following lemmas play dominant roˆ les in this subsection. Lemma 2.8 ([478, p. 48, Eq. (3)]) We have, for any test function ϕ, Z∞

xs−1 ϕ(x) dx =

0

Z1

 n ( j) (0) X ϕ xj  dx xs−1 ϕ(x) − j! 

j=0

0

Z∞ + 1

(4)

n X ϕ ( j) (0) xs−1 ϕ(x) dx + j! (s + j) j=0

(<(s) > −n − 1; s 6= −1, −2, . . . , −n; n ∈ N0 ). As an easy consequence of Lemma 2.8, one finds the following result. s−1 Lemma 2.9 ([478, p. 57]) The generalized function x+ / 0(s) has an analytic continuation to an entire function in s, such that s−1 x+ = δ (n) (x) (n ∈ N0 ), 0(s) s=−n

where δ is the delta function (see [478, Section 1.3]) and, for any test function ϕ, < δ (n) (x), ϕ(x) >:= (−1)n δ (n) (0)

(n ∈ N0 ).

216

Zeta and q-Zeta Functions and Associated Series and Integrals

Lemma 2.10 The generalized function f (x; s, u) =

u−1 (1 − x)s−1 + x+ 0(s) 0(u)

can be extended to an entire function in complex variables s and u. Proof. We need to show that, for any test function ϕ on [0, 1], the function of two complex variables s and u defined by Z1 0

(1 − x)s−1 xu−1 ϕ(x) dx 0(s) 0(u)

(<(u) > 0; <(s) > 0),

can be analytically continued to an entire function on all of C2 . This follows easily from Lemma 2.8, since the interval [0, 1] can be separated into [0, 1/2] ∪ [1/2, 1].  Remark 2 The Riemann zeta function ζ (s), defined by 2.3(1), can be obtained by the Mellin-transformation (see 2.3(4)): 1 ζ (s) = 0(s)

Z∞ 0

xs−2 ·x 1 dx = x e −1 s−1

Z∞ 0

xs−2 x · x dx 0(s − 1) e − 1

(<(s) > 1).

(5)

In view of Lemma 2.9, one finds that (s − 1) ζ (s) can be considered as the value of the generalized function xs−2 / 0(s − 1) (which is entire) on the test function x/(ex − 1). In this way, one immediately recovers the analytic continuation of ζ (s). Zhao [1260] used this idea to give an analytic continuation the multiple zeta functions of any depth as follows: Theorem 2.11 The multiple zeta function ζd (s1 , . . . , sd ) of depth d (d ∈ N) defined by (1), can be continued meromorphically to the whole space Cd with possible simple poles at sd = 1 and sd ( j) := sj + · · · + sd = d − j + 2 − ` ( j, ` ∈ N; 1 5 j < d). Proof. If d = 1, then ζ (s1 ) is the Riemann zeta function, defined by 2.3(1). So, one may assume d ∈ N \ {1}. By the identities X

e−n1 t1 −n2 t2 −···−nd td

0
=

d X

e−n1 t1 −(n1 +n2 )t2 −···−(n1 +···+nd )td

j=1, nj ∈N

=

d X Y j=1 nj ∈N

e−nj (tj +···+td ) =

d Y j=1


−1

etj +···+td − 1

The Zeta and Related Functions

217

and Z∞

e−α t t s−1 dt =

0

0(s) αs

(<(s) > 0; <(α) > 0),

(6)

it is easy to find that Z∞ Z∞ Z∞ s1 −1 s −1 t1 · · · tdd dt1 · · · dtd 0 (s1 ) · · · 0 (sd ) ζd (s1 , . . . , sd ) = ···
. Qd tj +···+td − 1 j=1 e 0

0

(7)

0

Now, put xd+1 := 0 and make the change of variables x1 · · · xj = tj + · · · + td , ( j ∈ N; 1 5 j 5 d).

if and only if tj = x1 · · · xj 1 − xj+1




(8)

It is not difficult to get the Jacobian of (8): ∂t1  ∂x1   . ∂(t1 , . . . , td ) = det   .. ∂(x1 , . . . , xd )   ∂td ∂x1 

 ∂t1 ∂t1 ··· ∂x2 ∂xd   .. ..  d−1 d−2 2  . · · · .  = x1 ·x2 · · · xd−2 ·xd−1 .  ∂td ∂td  ··· ∂x2 ∂xd

(9)

It follows from (8) and (9) that 0 (s1 ) · · · 0 (sd ) ζd (s1 , . . . , sd ) Z1 =

··· 0

Z1 Z∞ Y d 0

0

s ( j)−d+j−2

xj d

j=1

d Y

1 − xj


sj−1 −1

ϕ(x) dx1 · · · dxd ,

(10)

j=2

where the function ϕ(x) is given by 2 ·x x1d ·x2d−1 · · · xd−1 d , ϕ(x) := Qd x ···x j 1 − 1) j=1 (e

(11)

which is seen to be a test function in K. Letting uj := sd ( j) − d + j − 1, we can apply the generalized function
s −1 u −1 u1 −1 Y d 1 − xj +j−1 ·xj+j x1+

· 0 (u1 ) 0 s 0 u j−1 j j=2

218

Zeta and q-Zeta Functions and Associated Series and Integrals


Q on ϕ(x) and get exactly ( sd − 1) ζd (s1 , . . . , sd ) / dj =−11 0 uj , since 0 ( sd ) = (sd − 1) 0 (ud ). By Lemma 2.9 and Lemma 2.10, all of the following generalized functions

f1 (x; s) =

u1 −1 x1+

and fj (x; s) =

0 (u1 )


sj−1 −1

u −1

·xj+j

0 sj−1 0 uj

1 − xj

+

( j ∈ N; 2 5 j 5 d)

can be extended to entire functions in s. Furthermore, fj depends only on xj as a function of x. Hence, one can set the entire function ξ (s1 , . . . , sd ) :=

Y d

fj (x; s), ϕ(x)

 (12)

j=1

on Cd and define an analytic continuation of ζd (s1 , . . . , sd ) on Cd by Qd−1 ζd (s1 , . . . , sd ) :=

j=1

0 uj




sd − 1

ξ (s1 , . . . , sd ).

(13)

This completes the proof of the theorem, because Gamma functions have simple poles only at nonpositive integers (see 1.1(10) and 1.1(12)).  Zhao [1260, Theorem 6 and Theorem 7] evaluates the residues of ζd (s1 , . . . , sd ) at the simple poles given in Theorem 2.11 as follows: Theorem 2.12 The residue of ζd (s1 , . . . , sd ) at sd = 1 is 1 or ζd−1 (s1 , . . . , sd−1 ) , according to whether d = 1 or d ∈ N \ {1}. For depth d ∈ N \ {1} and i, ` ∈ N with 1 5 i 5 d − 1, the residue of ζd (s1 , . . . , sd ) on the hyperplane sd (i) = d − i + 2 − ` in Cd is equal to  ζi−1 (s1 , . . . , si−1 )

Baj 0 ad ( j) + uj

j=i+1

aj ! 0 ad ( j + 1) + uj + 1

 ad (i+1)=`−1 ai+1,...,a =0



d Y

X





,

(14)

d

where ζ (s0 ) := 1, ad ( j) := aj + · · · + ad , ad (d + 1) := 0, and uj := sd ( j) − d + j − 1. Proof. The case d = 1 is obvious, since ζ (s1 ) is the Riemann zeta function. So, we may assume d ∈ N \ {1}. Let ϕ(x) and ξ (s1 , . . . , sd ) be as in (11) and (12). In view

The Zeta and Related Functions

219

of (13), we find that Res ζd (s1 , . . . , sd ) = lim (sd − 1) ζd (s1 , . . . , sd ) sd →1     d−1  Y
 = lim 0 uj lim ξ (s1 , . . . , sd ) , sd →1  sd →1

sd =1

j=1

where the two involved limits are readily seen to be lim

sd →1

d−1 Y

d−2 Y
0 uj = 0 (sd−1 − 1) · 0 (sd−1 ( j) − d + j)

j=1

j=1

and ξ (s1 , . . . , sd−1 ) . 0 (sd−1 )

lim ξ (s1 , . . . , sd ) =

sd →1

Now, it is easy to see that Qd−2 Res ζd (s1 , . . . , sd ) =

sd =1

j=1

0 (sd−1 ( j) − d + j)

sd−1 − 1 ·ξ (s1 , . . . , sd−1 ) = ζd−1 (s1 , . . . , sd−1 ).

The proof of the remaining part, that is (14), is left to an interested reader.



Remark 3 By the power series definition, one easily sees that ζ2 (0, s2 ) = ζ (s2 − 1) − ζ (s2 ),

(15)

whereas it is known (see 3.3(35)) that ζ2 (s1 , s2 ) + ζ2 (s2 , s1 ) + ζ (s1 + s2 ) = ζ (s1 ) ζ (s2 ).

(16)

Setting s2 = 0 in (16) and using (15) yields ζ2 (s1 , 0) = ζ (s1 ) ζ (0) − ζ (s1 − 1).

(17)

Now, setting s2 = 0 in (15) and s1 = 0 in (17), together with 2.3(10), gives lim lim ζ2 (s1 , s2 ) =

s2 →0 s1 →0

5 12

and

1 lim lim ζ2 (s1 , s2 ) = . s1 →0 s2 →0 3

(18)

It is observed that the limits of a function at some point along different routes in the multivariable space may be different.

220

Zeta and q-Zeta Functions and Associated Series and Integrals

From (10) and the identity (use 1.7(2))    1 x2 y 1  2 2 = 1 − (1 + y)x + y + 3y + 1 x + O x3 2 12 (ex − 1) (exy − 1)

(x → 0), (19)

we readily obtain (see Zhao [1260, p. 1280]) that ζ2 (s1 , s2 ) =

4 s1 + 5 s2 + R (s1 , s2 ), 12 (s1 + s2 )

(20)

where R (s1 , s2 ) represents an analytic function of (s1 , s2 ) in a neighborhood of (0, 0) and lim(s1 ,s2 )→(0,0) R (s1 , s2 ) = 0. It is seen that (20) also gives the same limit values as in (18).

Euler-Maclaurin Summation Formula We begin by remarking that most of this subsection is due to Akiyama et al. [16] and recalling another version of Euler-Maclaurin summation formula (cf. [3, p. 806] and 1.3(68)): Let N ∈ N and a, b ∈ R with a < b. Suppose that f ∈ CN ([a, b]) denotes (as usual) the set of functions having continuous derivatives of Nth order on the interval [a, b]. Then, we have

X

f (n) =

a
Zb

K X (−1)k e Bk (b)f (k−1) (b) f (x)dx + k! k=1

a

−

K X (−1)k

Zb

k=1

a

(−1)K e Bk (a)f (k−1) (a) − k! K!

(21) e BK (x)f (K) (x)dx,

where e Bk (x) is the periodic Bernoulli polynomial, defined by e Bk (x) := Bk (x − [x]) ([x] denotes the greatest integer 5 x). Setting a = 1, b = m and K = ` + 1 in (21) yields m X n=1

f (n) =

Zm

f (x) dx +

1

(−1)`+1 − (` + 1)!

`  X 1 Bk+1  (k) f (m) − f (k) (1) (f (1) + f (m)) + 2 (k + 1)! k=1

Zm

(22) e B`+1 (x) f (`+1) (x) dx,

1

where f ∈ C`+1 ([1, m]), m ∈ N and ` ∈ N0 .

The Zeta and Related Functions

221

f (x) = x−sPin (22). We get, by subtracting the resulting identity of PmConsider −s −s = ζ (s), the following formula: from that of ∞ n=1 n n=1 n ( ) m ` X X 1 (s)k ak 1 m1−s − 1 1 φ` (m, s) = − − + ζ (s) − , + ns 1−s 2 ms s−1 ms+k n=1

(23)

k=1

where (s)k is the Pochhammer symbol given in 1.1(5), ak := Bk+1 /(k + 1)!, and Z∞ (s)`+1 e B`+1 (x) x−s−`−1 dx (` + 1)! m   −<(s)−` =O m ,

φ` (m, s) : =

(24)

where, for example, see Hardy [537, p. 332, 13.10.3]. We rewrite (23) in the following equivalent form: ∞ X n=m+1

`

X (s)k ak 1 m1−s 1 = −φ (m, s) + − + ` s s n s − 1 2m ms+k

(<(s) > 1).

(25)

k=1

Consider the following multiple Zeta function in two variables: ζ2 (s1 , s2 ) =

X

1

ns1 ns2 0
(<(si ) > 1; i = 1, 2).

We make use of (25) to get ζ2 (s1 , s2 ) =

∞ X 1 ns11

n1 =1

=

∞ X n1 =1

1 ns11

∞ X n2 =n1 +1

1 ns22 `

X (s2 )k ak n1−s2 1 − s2 + −φ` (n1 , s2 ) + 1 s2 +k s2 − 1 2 n1 k=1 n1

(

ζ (s1 + s2 − 1) ζ (s1 + s2 ) = − s2 − 1 2 ` ∞ X X φ` (n1 , s2 ) + (s2 )k ak ζ (s1 + s2 + k) − ns11 k=1

) (26)

n1 =1

for <(si ) > 1 (i = 1, 2). The terms on the right hand side, except the last one, have meromorphic continuations. The last sum is absolutely convergent and, hence, holomorphic in <(s1 + s2 + ` − 1) > 0. Thus, we now have a meromorphic continuation of ζ2 (s1 , s2 ) to C2 , which is holomorphic in n o / {2, 1, 0, −2, −4, −6, . . .} . (s1 , s2 ) ∈ C2 | s2 6= 1, s1 + s2 ∈

222

Zeta and q-Zeta Functions and Associated Series and Integrals

It is easy to see that this manipulation can be employed to a multiple zeta function with d variables. Indeed, we have ζd (s1 , . . . , sd ) =

∞ X 1 ns11

n1 =1

=

∞ X n1 =1

1 ns11

∞ X n2 =n1 +1 ∞ X n2 =n1 +1

1 ··· ns22 1 ··· ns22

∞ X

∞ X

∞ X

1 `

X (sd )k ak 1 · −φ` (nd−1 , sd ) + − sd + sd +k sd − 1 2 nd−1 k=1 nd−1 =

1 s ndd

s n d−1 nd−1 =nd−2 +1 d−1

1−s

(

1

s n d−1 nd−1 =nd−2 +1 d−1 nd =nd−1 +1

nd−1d

) (27)

ζd−1 (s1 , . . . , sd−2 , sd−1 + sd − 1) ζd−1 (s1 , . . . , sd−2 , sd−1 + sd ) − sd − 1 2 ` X + (sd )k ak ζd−1 (s1 , . . . , sd−2 , sd−1 + sd + k) k=1

X

−

0
φ` (nd−1 , sd ) sd−1 ns11 · · · nd−1

(<(si ) > 1; i = 1, . . . , d).

Since X 0
φ (n , s ) X n−`−<(sd )+d−2 ` d−1 d d−1 s1 5 d−1 n1 · · · nsd−1 nLd−1 nd−1

with L := <(sd−1 ) +

X

<(si ),

15i5d−2 <(si )50

the last summation is absolutely convergent in the set X ` − d + 1 + <(sd ) + <(sd−1 ) + <(si ) > 0.

(28)

15i5d−2 <(si )50

Since ` can be taken arbitrarily large, we get an analytic continuation of ζd (s1 , . . . , sd ) to Cd . Now, we consider the set singularities. It is seen that the singular part of ζ2 (s1 , s2 ) may be formally given as follows: ζ (s1 + s2 − 1) X ak1 (s2 )k1 + , s2 − 1 s1 + s2 + k1 − 1 k1 =0

The Zeta and Related Functions

223

since, by analytic continuation, ζ (s) can be written as ζ (s) = 1/(s − 1) + τ (s), where τ (s) is an entire function. We find from this expression that s2 = 1,

s1 + s2 ∈ {2, 1, 0, −2, −4, −6, . . .}

forms the set of all singularities of ζ2 (s1 , s2 ). For the case ζ3 (s1 , s2 , s3 ), by using the singular part of ζ2 , we see that singularities lie on s3 = 1,

s2 + s3 ∈ {2, 1, 0, −2, −4, −6, . . .}

and s1 + s2 + s3 ∈ {3, 2, 1, 0, −1, −2, −3, . . .}. To show these are all singularities of ζ3 (s1 , s2 , s3 ), it suffices to prove that no singularities, defined by one of the above equations, will identically vanish. This can be shown by changing variables: u1 = s1 ,

u2 = s2 + s3 ,

u3 = s3 .

In fact, it is seen that the singular part of ζ3 (u1 , u2 − u3 , u3 ) is given by X 1 ζ2 (u1 , u2 − 1) + ak2 (u3 )k2 ζ2 (u1 , u2 + k2 ). u3 − 1 k2 =0

It is found from this expression that the singularities of ζ2 (u1 , u2 + k2 ) are summed with functions of u3 of different degrees. Thus, these singularities, as a weighted sum by another variable u3 , will not vanish identically. Similarly, we can prove Theorem 2.13 below. Theorem 2.13 The multiple zeta function ζd (s1 , . . . , sd ) continues meromorphically to Cd and has singularities on sd = 1,

sd−1 + sd = 2, 1, 0, −2, −4, . . . ,

and j X

sd−i+1 ∈ Z5j

( j = 3, 4, . . . , d),

i=1

where Z5j is the set of integers less than or equal to j. We conclude this section by remarking the following facts: Akiyama et al. [16] defines the multiple zeta values at nonpositive integers by
ζd (−r1 , . . . , −rd ) := lim · · · lim ζd (s1 , . . . , sd ) rj ∈ N0 ; j = 1, . . . , d . s1 →−r1

sd →−rd

(29)

224

Zeta and q-Zeta Functions and Associated Series and Integrals

If, for simplicity, we write (s)−1 :=

1 , s−1

then it follows from (24), (27) and (29) that ζd (−r1 , . . . , −rd ) =

rd X

(−rd )k ak ζd−1 (−r1 , . . . , −rd−2 , −rd−1 − rd + k), (30)

k=−1

where we have used the fact that, in view of (24), φ` (nd−1 , −rd ) = 0

(` = rd ).

Akiyama et al. [16] and Akiyama and Tanigawa [17] made a basic use of (30) to give the multiple zeta values at nonpositive integers as asserted by Theorem 2.13 below. Theorem 2.14 [17, p. 330, Theorem 2] ζd (−n, 0, . . . , 0) =

d j Bn+j (−1)n X s(d, j) d! n+j

(n ∈ N0 ),

(31)

j=1

where s(d, j) denotes the Stirling numbers of the first kind (see Section 1.6) and Bj the Bernoulli numbers (see Section 1.7).

Problems 1. Show that     p 1 p q−2k [ψ(2k) − log(2π)] B2k ζ 0 −2k + 1, = [ψ(2k) − log(2π q)] B2k − q 2k q 2k     q−1 2π pn (−1)k+1 π X (2k−1) n + sin ψ q q (2π q)2k n=1

+

q−1 (−1)k+1 2 · (2k − 1)! X

(2π q)2k

ζ 0 (−2k + 1) + q2k

n=1

 cos

2π pn q



  n ζ 0 2k, q

(p < q; p, q, k ∈ N).

(Miller and Adamchik [829, p. 203, Proposition 1]) 2. Prove that there exist constants Dk , defined by ! n X k log Dk := lim m log m − p(n, k) (k ∈ N0 ), n→∞

m=1

The Zeta and Related Functions

225

where the definition of p(n, k) in Adamchik [6, p. 198, Eq. (20)] is corrected here as follows:   nk+1 1 nk log n − p(n, k) := log n + 2 k+1 k+1   j k k−j X
X n Bj+1 1 log n + 1 − δkj  + k! ( j + 1)! (k − j)! k−`+1 `=1

j=1

and δkj is the Kronecker symbol defined by 1.6(25). (Bendersky [114, pp. 273–275]; Adamchik [6, p. 198]) 3. For the constants Dk (k ∈ N0 ), defined in Problem 2, show that 1

D0 = (2π ) 2

D1 = A,

D2 = B and

D3 = C

and log Dk =

Bk+1 Hk − ζ 0 (−k) k+1

(k ∈ N0 ),

where Bn are the Bernoulli numbers and Hn are the harmonic numbers, and the mathematical constants A, B, and C are given as in Section 1.4. (Adamchik [6, pp. 198–199]) 4. Prove that Zπ  h  p i3 h    i log 2 sin 12 t 2 cos 12 t dt = −2π ζ (3) 43 1 + p3 − 83 p + p2 . 0

(Lewin [752, p. 220]) 5. The Barnes double Zeta function ζ2 (v; α, w) is the meromorphic continuation of the series defined by ζ2 (v; α, w) :=

∞ X ∞ X

(α + m + nw)−v

(α > 0; w > 0; v ∈ C).

m=0 n=0

The functions ρ2 (1, w) and 02 (α, (1, w)) are defined by   log ρ2 (1, w) := − lim ζ20 (0; α, w) + log α α→0

and   02 (α, (1, w)) := ρ2 (1, w) exp ζ20 (0; α, w) , respectively. (a) Prove that, for any N ∈ N and <(v) > −N + 1 (v ∈ C \ [Z− 0 ∪ {1, 2}]), ζ (v − 1) 1−v w v−1 N−1 X −v + ζ (−n, α)ζ (v + n)w−v−n + RN (v; α, w) n

ζ2 (v; α, w) = ζ (v, α) +

n=0

226

Zeta and q-Zeta Functions and Associated Series and Integrals

with the estimate:   RN (v; α, w) = O w−<(v)−N , where the O-constant depends on v, N and α. (b) Show also that, for N ∈ N \ {1, 2},   1 1 ζ20 (0; α, w) = − w log w + − ζ 0 (−1) w 12 12     1 3 1 1 − α log w + log 0(α) + α− log(2π) + 2 2 2 4 + ζ (−1, α) w−1 (log w − γ ) +

N−1 X n=2

(−1)n ζ (−n, α) ζ (n) w−n + R0N (0; α, w) n

and   1 log 02 (α, (1, w)) = α log 2πw−1 + log 0(α) 2 + [ζ (−1, α) − ζ (−1)] w−1 (log w − γ ) +

N−1 X n=2

(−1)n [ζ (−n, α) − ζ (n)] ζ (n) w−n + SN (α, (1, w)) n

with the estimates: h i R0N (0; α, w) = O w−N (| log w| + 1) and  i SN (α, (1, w)) = O w−N (| log w| + 1) , where the O-constants depend on N and α and the prime in ζ20 (v; α, w) denotes differentiation with respect to v. (Matsumoto [804, 805]) 6. For the Bernoulli numbers and polynomials, show that (a) B2n = 4n (−1)n+1

Z∞

x2n−1 e2πx − 1

0

dx

(n ∈ N);

(b) |B2n (x)| ≤ |B2n |

(n ∈ N; 0 5 x 5 1).

7. Show that ζ (s) =

Z∞

1 1 − 21−s




0(s)

0

xs−1 dx ex + 1

(<(s) > 0; s 6= 1). (See, e.g., Chen [240, p. 263])

The Zeta and Related Functions

227

8. Prove that m+n+ 12

Z

log 0(t) dt = −

m−1 X

k log k +

k=1

m

m+n−1 1 X (2k + 1) log (2k + 1) 2 k=1

1 m(m − 1) 3 0 − (m + n)2 (1 + log 2) + − ζ (−1) 2 2    2  √ 3 1 1 + − + log 2 ζ (−1)+ n + log 2π (m ∈ N; n ∈ N0 ). 2 2 2 (Elizalde et al. [412, p. 19]) 9. Prove that Zz

tn ψ(t) dt = (−1)n−1 ζ 0 (−n) +

0

−

n X

(−1)k

k=0

(−1)n Bn+1 Hn n+1

    n−k n X n n−k 0 n z Bk+1 (z) Hk + (−1)k z ζ (−k, z) k k k+1 k=0

(n ∈ N0 ; <(z) > 0). (Adamchik [6, p. 197]) 10. Prove that    ∞  za 1 s−1 X s − 1 ζ (s − 2k, a) 8(z, s, a) = log + (log z)2k 0(1 − s) z 2k 0(2k + 1 − s) k=0  ∞  X s−1 ζ (s − 2k − 1, a) (| log z| < 2π ; s 6∈ N; a 6∈ Z− − (log z)2k+1 0 ). 2k + 1 0(2k + 2 − s) k=0

Also deduce the relatively more familiar result 2.5(13). 11. Prove that   4(−1)n 2n E2n−1 (0) = (2n − 1)! 2 − 1 ζ (2n) (n ∈ N), (2π )2n where En (x) denotes the Euler polynomials (see Section 1.6). (Cf. Srivastava [1084, p. 390]) 12. Let f : N × N → R be a function. For a fixed n ∈ N, let Tn := {(k, j) ∈ N × N | 1 5 k, j 5 n; k + j = n + 1} and Sn :=

X

f (x, y),

(x,y)∈Tn

the sum being taken over all pairs (x, y) ∈ Tn . Show that Sn =

n X k=1

f (k, k) +

n X k−1 X

{ f ( j, k) + f (k, j) − f ( j, k − j)}.

k=2 j=1

(Sitaramachandrarao and Sivaramsarma [1035, p. 603])

228

Zeta and q-Zeta Functions and Associated Series and Integrals

13. Prove that  2

∞ X q=1

1   qn 

 q−1 X

n−2

k=1 (k,q)=1

X 1 1  = n+2− ζ (n − j) ζ ( j + 1)  k ζ (n + 1)

(n ∈ N \ {1}),

j=1

where an empty sum is understood (as usual) to be nil and (k, q) = 1 denotes that k and q are relatively prime. (Cf. Equation (54); see also Sitaramachandrarao and Sivaramsarma [1035, p. 602]) 14. Prove that 2n ∞ X 1X Hk (−1)j ζ ( j) ζ (2n − j + 2) = 2 k2n+1

(n ∈ N),

j=2

k=1

and ∞ n−1 X Hk  n 1X = 1 + ζ (n + 1) − ζ ( j) ζ (n − j + 1) kn 2 2

(n ∈ N \ {1}),

j=2

k=1

where Hk are the harmonic numbers defined by 3.2(36). (Georghiou and Philippou [479, p. 29]) 15. For bounded maps f , g : N → C, let Sf ,g (u, v; w) =

∞ X f (r + k) g(k) ru ku (r + k)w

u, v ∈ N; w ∈ R+




r,k=1

and Cf ,g (x, y) =

r−1 ∞ X f (r) X g(k) + g(r − k) rx ky r=2

(x > 1; y ∈ N).

k=1

Show that Sf ,g (u, v; w) + Sf ,g (v, u; w) =

 u−1  X v+j−1 v−1

j=0

+

Cf ,g (v + w + j, u − j)

 v−1  X u+j−1 j=0

u−1

Cf ,g (u + w + j, v − j).

(Subbarao and Sitaramachandrarao [1133, p. 246]) 16. Prove that ∞ X (−1)r+k 1 = ζ (3) rk (r + k) 4

r,k=1

and ∞ X (−1)k−1 5 = ζ (3). rk (r + k) 8

r,k=1

(Subbarao and Sitaramachandrarao [1133, p. 247])

The Zeta and Related Functions

229

17. Prove that (2) ∞ X Hk (n + 2)(2n + 1) = ζ (2) ζ (2n + 1) − ζ (2n + 3) 2 k2n+1 k=1

+2

n+1 X

( j − 1)ζ (2j − 1) ζ (2n + 4 − 2j)

(n ∈ N),

j=2 (m)

where Hn

(m)

H0

denotes the generalized harmonic number, defined by

=0

Hn(m) =

and

n X

`−m

(m, n ∈ N).

`=1

(Georghiou and Philippou [479, p. 35]) 18. Prove that ∞ X m, n=0

1 13 π 8
= 28350 m2 + n2

0

m6

∞ X

and

m, n=0

0

m2

1
= m2 − mn + n2

√

3π4 , 30

where the prime denotes that the summations are taken over all pairs of integers, except (0, 0). (Smart [1042, p. 10]) 19. Prove that ∞ X 1 π2 . (H2k−1 − Hk − log 2) = (log 2)2 − k 6 k=1

(Knuth [679, p. 138]) 20. Let a, b, c ∈ R with a > 0 and d = b2 − 4ac < 0. The Epstein Zeta function is given by Z(s) :=

−s 1 X0  2 a m + b mn + c n2 2

(<(s) > 1),

where the prime denotes that the summation is to be taken over all pairs (m, n) of integers other than the pair (0, 0). Let k ∈ R+ be a number satisfying  2 c b k = − a 2a 2

(a, b, c ∈ R; a > 0)

and suppose that the Bessel function Kν (z) is defined by Kν (z) =

1 2

Z∞

z

e− 2

u+ u1




uν−1 du =

0

Z∞ = 0

e−z cosh t cosh(νt) dt

1 2

Z∞

e−z cosh t eνt dt

−∞

 π ν ∈ C; | arg z| < . 2

230

Zeta and q-Zeta Functions and Associated Series and Integrals

Also, let σν (n) :=

X

dν =

d|n

X  n ν d|n

d

(ν ∈ C; n ∈ N).

Show that as Z(s) = ζ (2s) + k1−2s ζ (2s − 1)

    0 s − 12 0 12 0 (s)

+

π s 1 −s k 2 H(s) 0(s)

(<(s) > 1),

where H(s) = 4

∞ X

1

ns− 2 σ1−2s (n) cos

n=1



nπ b a



Ks− 1 (2π kn). 2

Show, also, that H(s) is an entire function of s, such that H(s) = H(1 − s). (Bateman and Grosswald [101, p. 366]) 21. Show that  o 1n az + b , −2n = A(z, −2n) + 1 − (cz + d)2n ζ (2n + 1) (cz + d) A cz + d 2       c 2n+2 (−1)n (2π )2n+1 i X X 2n + 2 j jd + Bk B¯ 2n−k+2 {−(cz + d)}k−1 2 (2n + 2)! k c c 2n



j=1 k=0

(c, n ∈ N), where A(z, −2n) denotes a Lambert series in the variable e2πiz , defined by A(z, −2n) :=

∞ X

k−2n−1

k=1

e2π ikz 1 − e2πikz

and B¯ n (x) := Bn (x − [x]) ([x] being, as usual, the greatest integer in x). (Cf. Berndt [119, p. 505]; see also Apostol [57, p. 153]) 22. Show that     X   k `−1 X a a+1 k (−1)j (` − 1 − j + a)−s 2k−s ζ s, − ζ s, = (−1)`−1 j 2 2 `=1

+

∞ X n=0

(−1)n+k

k X j=0

j=0

  k (−1)j (n + k − j + a)−s j

(k ∈ N; <(s) > −k),

where ζ (s, a) (a > 0) is the Hurwitz (or generalized) Zeta function, defined by 2.2(1). (Balakrishnan [91, p. 205])

The Zeta and Related Functions

231

23. Show that π

 2m−2k−2 Z 3 m−2 X (−1)k 2m − 2  θ 2k+1  θ dθ log 2 sin 2k + 1 2k 2 2 k=0 0 "  #   (−)m π 2m 1 2m−1 1 = − 2 1 − 2m−1 B2m (m ∈ N \ {1}) 8m(2m − 1) 6 2 and π

Z 3 m−1 X

(−1)k

k=0

0

=

   2k   2m−2k 2m θ θ log 2 sin dθ 2k 2 2

  (−)m π 2m+1 1 E − 2m 22m+2 (2m + 1) 32m

(m ∈ N),

where Bn and En are the Bernoulli and Euler numbers, respectively, given in Section 1.7. (Zhang and Williams [1253, p. 282]) 24. Show that π

∞ X

1

n=1

nm+2

(−1)m 2m =
2n m! n

Z3

θ θ log 2 sin 2 



m dθ

(m ∈ N0 )

0

and 2Zlog τ   m ∞ X (−1)m 2m θ (−1)n−1 = θ log 2 sinh dθ
m+2 2n m! 2 n n=1 n

(m ∈ N0 ).

0

(Zhang and Williams [1253, pp. 286 and 288]) 25. Show that π Z 3 "

0

 4   2 # θ 3θ2 θ 253 π 5 log 2 sin − log 2 sin dθ = 3 4 2 2 2 2 ·3 ·5

and π 4  # Z3 "   θ θ3 θ 313 π 6 θ log 2 sin − log 2 sin dθ = 4 6 . 2 2 2 2 ·3 ·5·7

0

(Zucker [1267, pp. 98 and 99]) 26. Prove each of the following claims: (a) For all a1 , a2 , . . . ∞ X k=1

a1 a2 · · · ak−1 1 = . x (x + a1 ) · · · (x + ak )

232

Zeta and q-Zeta Functions and Associated Series and Integrals

(b) ζ (3) =:

∞ ∞ X 5 X (−1)n−1 1 =
. 3 2n 2 n3 n=1 n n n=1

(c) Consider the recursion:   n3 un + (n − 1)3 un−2 = 34 n3 − 51 n2 + 27 n − 5 un−1

(n ∈ N \ {1}).

Let {bn } be the sequence defined by b0 = 1, b1 = 5 and bn = un for all n ∈ N \ {1}; then the bn are all integers. Let {an } be the sequence defined by a0 = 0, a1 = 6 and an = un for all n ∈ N \ {1}; then the an are rational numbers with denominator dividing 2[1, 2, . . . , n]3 (here, [1, 2, . . . , n] is the lowest common multiple of 1, . . . , n). (d) an /bn → ζ (3) as n → ∞; indeed, the convergence is so fast as to prove that ζ (3) cannot be rational. To be precise, for all integers p, q with q sufficiently large relative to  > 0, ζ (3) − p > 1 q qθ +

and

θ = 13.41782 · · · ,

which proves the irrationality of ζ (3). (Cf. van der Poorten [1180, p. 195]; see also Ap´ery [56]) 27. Prove the following inequalities: For n, p ∈ N, ! n !   n p−1 X n X X 1 1 X 1 8 1 0 ≤ p+ − < 2 n k2p k2p−2j k2j k=1

j=1

k=1

k=1

and ! X n k+1 2 X n 1 (−1) < 6. −2 n 2 2k − 1 (2k − 1) k=1 k=1 (Hovstad [571, p. 93]) 28. Show that the first inequality of Problem 27 implies that ∞ ∞ X X 1 1 = R p k2p k2 k=1

!p (p ∈ N),

k=1

where Rp is a rational number, satisfying the recurrence formula:

R1 = 1

and

  p−1 X 1 p+ Rp = Rj Rp−j 2

(p ∈ N \ {1}).

j=1

(Hovstad [571, p. 93])

The Zeta and Related Functions

233

29. Prove that ζ (s) has the factorization ζ (s) =

  Y  s s ebs   1− eρ , 1 ρ 2 (s − 1) 0 1 + 2 s ρ

where b = −1 −

γ + log(2π ); 2

γ is the Euler-Mascheroni constant, and the product is taken over all the so-called nontrivial zeros of ζ (s). (Cf. Titchmarsh [1151, pp. 30–31]; see also Melzak [821, p. 111]) 30. Prove that   X   ∞ 1 (2n)! 1 ζ ,a + 1 = ζ n + (−a)n (|a| < 1) 2 2 22n (n!)2 n=0

and ζ



  n−1  1 X 1 k 1 , na = n− 2 ,a + ζ 2 2 n

(n ∈ N).

k=0

(Powell [911, p. 117]) 31. Let (s − 1) ζ (s, a) = 1 +

∞ X

γn (a) (s − 1)n+1

(0 < a ≤ 1; n ∈ N0 ).

n=0

Show that ! m X {log(k + a)}n {log(m + a)}n+1 (−1)n lim − γn (a) = , n! m→∞ k+a n+1 k=0

which, for a = 1, gives the coefficients in the Laurent expansion of ζ (s) about s = 1 (cf. Equation 2.3(7)). (Berndt [118, p. 152]) 32. Prove that lim sup t→∞

|ζ (1 + it)| ≥ eγ , log (log t)

where γ is the Euler-Mascheroni constant defined by 1.1(3). (Titchmarsh [1150, p. 79]) 33. Prove that ∞ X
zn Lim ez = ζ (m − n) n! n=0 n6=m−1

  m−1 1 z 1 + 1 + + ··· + − log(−z) 2 m−1 (m − 1)!

(m ∈ N; |z| < 2π). (Cohen et al. [336, p. 26])

234

Zeta and q-Zeta Functions and Associated Series and Integrals

34. Prove that ∞ n−1 4 1 5X X 1 − ζ (5) = 2 k 2 5 n2 n=1

!

k=1

(−1)n
. n3 2n n

(van der Poorten [1181, p. 274]) 35. Let K be a normal extension field of degree n over the rational number field Q. Denote by OK the integer ring of K. Let I(K) be the set of all nonzero ideals of OK , Na the absolute norm of an ideal a ∈ I(K) and Tr α the trace of α ∈ K over Q. We assume that [K : Q] = n > 1, let OK,0 = {α ∈ OK | Tr α = 0}; TK = min{Tr α | α ∈ OK , Tr α > 0}. For a ∈ I(K) and a0 = OK,0 ∩ a, define  T(a) = Min

 Trα | α ∈ a, Trα > 0 ; T(K)

N0 (a) = #{OK,0 /a0 }, where OK,0 /a0 is the quotient of Z-module OK,0 by submodule a0 . For any x ∈ R+ , let jK (x) = #{a ∈ I(K) | Na ≤ x}; jK,0 (x) = #{a ∈ I(K) | N0 (a) ≤ x}   Na = # a ∈ I(K) | ≤x , T(a) and q ≡ 1 (mod n). Suppose that K is the subfield of the qth cyclotomic field Cq , such that K/Q is a tamely ramified cyclic extension of degree n. Show that ζ (2k + 1) =

2 qn − qn−1 − q jk (x) ζ (2k) lim . x→+∞ jK,0 (x) qn − 1 (Lan [728, p. 273])

36. Prove that ∞ ∞ (2) X 469 3 (Hn )3 11 X Hn − = ζ (8) − 16 ζ (3) ζ (5) + ζ (2) {ζ (3)}2 5 4 32 2 n n6 n=1

n=1

and ∞ ∞ (2) X 561 47 (Hn )3 13 X Hn − = ζ (10) − {ζ (5)}2 7 8 4 20 4 n n n=1

n=1

49 15 − ζ (7) ζ (3) + 3 ζ (2) ζ (3) ζ (5) + {ζ (3)}2 ζ (4). 2 4 (Flajolet and Salvy [454, p. 27])

The Zeta and Related Functions

235

37. Prove that, for an odd weight m = p + q,      ∞ X 1 (−1)p m − 1 (−1)p m − 1 H (p) (n) − = ζ (m) − p q nq 2 2 2 n=1

+

[p/2] X m − 2k − 1 1 − (−1)p ζ (p) ζ (q) + (−1)p ζ (2k) ζ (m − 2k) q−1 2 k=1

+ (−1)p

[q/2] X k=1



m − 2k − 1 ζ (2k) ζ (m − 2k), p−1

where ζ (1) should be interpreted as 0 wherever it occurs. (Cf. Borwein et al. [146, p. 278]; see also Flajolet and Salvy [454, p. 22]) 38. Prove that ∞ 2n−1 X X (−1)n+k 27 = πG− ζ (3) 2 16 n k n=1 k=1

and ∞ 2n−1 X X (−1)n−1 29 ζ (3), = πG− 16 n2 k n=1 k=1

where G denotes the Catalan constant defined by 1.3(16). (Sitaramachandrarao [1034, p. 13]) 39. Prove that, for m ∈ N and <(a) > 0, ∂ ζ 0 (−m, a) = ζ (z, a) ∂z z=−m 1 1 1 = am+1 log a − am+1 − am log a m+1 2 (m + 1)2 ∞ 1 m−1 X m m−1 a log a + a + α2k a−(2k−m+1) , + 12 12 k=1

where

α2k :=

     2k    j X  B m m (−1) 2k+2     log a +    2k + 2 2k + 1 j 2k − j + 1

(2k 5 m − 1),

j=0

m     B2k+2 X m (−1)j      2k + 2 j 2k − j + 1

(2k = m).

j=0

(Elizalde [410, p. 349]) 40. Define constants 8k , for k > 1 an odd integer, by 8k = −

k−2 d d−1 π 2 X (−1) 2 ζ (k − d + 1). π d! d=1 d odd

236

Zeta and q-Zeta Functions and Associated Series and Integrals

Show that, for real r ≥ 1 and s > 1, ∞ k−1   X X 1 k ζ (r, s) = − ζ (r + s) + 8k η(r − j) η(s − k + j), 2 j j=0 j even

k=3 k odd

where the Eta function is defined by   η(s) := 1 − 21−s ζ (s). (Crandall and Buhler [344, p. 279]) 41. Let f (s) be a function defined by the Dirichlet series as follows: f (s) :=

∞ X

n−s

n=2

X

k−1

(<(s) = σ > 1).

k
Show that f (s) is analytic in the whole complex s-plane, except at simple poles s = 0 and s = 1 − 2a (a ∈ N) with residues given by Res f (s) = − s=0

1 2

and Res f (s) = −

s=1−2a

B2a 2a

(a ∈ N),

and a pole of order 2 at s = 1 with residue given by Res f (s) = γ , s=1

where B2a are the Bernoulli numbers, defined by 1.6(2), and γ is the Euler-Mascheroni constant, defined by 1.1(3). (Matsuoka [813, p. 399]) 42. By using the definitions and restrictions as in Problem 20 of Chapter 2 with an additional condition c > 0, show that Z(s) = ζ (2s) a−s +

22s−1 as−1

√

π 1

0(s) (−d)s− 2

  1 ζ (2s − 1) 0 s − + Q(s), 2

where 1

2π s ·2s− 2

∞ X



nπb Q(s) = √ n σ1−2s (n) cos s 1 − a a 0(s)·(−d) 2 4 n=1 ! √ Z∞  3 πn −d  u + u−1 du. · us− 2 exp − 2a s− 12



0

(Selberg and Chowla [1017, p. 87])

The Zeta and Related Functions

237

43. Let r, s ∈ N0 with r > s. Show that Z1 Z1 0 0

x r ys dxdy 1 − xy

is a rational number whose denominator is a divisor of dr2 , where dr denotes the lowest common multiple of 1, 2, . . . , r. (Beukers [126, p. 268]) 44. Prove that ζ (3) =

∞ 1X 56 n2 − 32 n + 5 (−1)n−1 4 (2n − 1)2 n=1

1 3n
2n
3 n n n

and ζ (3) =

∞ X n=0

(−1)n 5265 n4 + 13878 n3 + 13761 n2 + 6120 n + +1040 .
3n
(4n + 1)(4n + 3)(n + 1)(3n + 1)2 (3n + 2)2 72 4n n n (Amdeberhan [34, p. 2])

45. Prove that     n ∞ X (−1)n X 1 1 1 4 = − 24 Li + 21 ζ (3) log 2 + (log 2) 4 6 2 n2 k2 n=1

k=1

+

17 4 π2 (log 2)2 + π , 6 480

where Li4 (z) denotes the Tetralogarithm (see Section 2.4). (Daud´e et al. [367, p. 421]) 46. Prove that   h    i 2 (log(2n + 1)) π (−1)n 2 ζ 00 0, 14 − ζ 00 0, 34 = 2n + 1 4 n=1     2 0 41 π −4[γ + log(2π )] log     + [γ + log(2π)]2 + 12  0 43

∞ X

and Z∞ 0

  i (log t)2 π  h 00  1  dt = 2 ζ 0, 4 − ζ 00 0, 34 cosh t 2      0 14 1 2 2 −4 log(2π ) log     + [log(2π)] + π . 4  0 34 (Shail [1021])

238

Zeta and q-Zeta Functions and Associated Series and Integrals

47. For an integer a = −1, prove the following asymptotic expansion: Ga (x) :=

∞ X

ζ (n − a)

n=a+1

(−x)n n!

(−x)a+1 {log x − ψ(a + 2) + γ } (a + 1)! !   a n X X (a−k)! 1 1 n + (−x) − +O (a+1)! (n−k)! n! (a + 1 − n) x

=−

n=0

(x → ∞).

n=0

(Buschman and Srivastava [195, p. 296]) 48. For any multi-index k = (k1 , k2 , . . . , kr )) (ki ∈ N), the weight wt(k) and depth dep(k) of k are defined by |k| =k1 + k2 + · · · + kr and r, respectively. The height of the index k is also defined by ht k = # j | kj = 2 . Denote, by I(k, r), the set of multi-indices k of weight k and depth r and, by I0 (k, r), the subset of I(k, r) with admissible indices, that is indices with the additional requirement that k1 = 2. For (k1 , . . . , kr ) ∈ I0 (k, r), the multiple zeta value (MZV) and the non-strict multiple zeta value (MZSV) can often be defined, respectively, as follows: ζ (k1 , k2 , . . . , kr ) :=

X

1

k1 kr n1 >···>nr >0 n1 · · · nr

and ζ ∗ (k1 , k2 , . . . , kr ) :=

X

1

nk1 · · · nkr r n1 =···=nr =1 1

.

Prove the following formulas: (a) Sum Formula. For r < k (r, k ∈ N), there hold X

ζ (k) = ζ (k) and

k∈I0 (k, r)

X

ζ ∗ (k) =

k∈I0 (k, r)

  k−1 ζ (k). r−1

(b) Cyclic Sum Formula. For (k1 , . . . , kr ) ∈ I0 (k, r), r kX i −2 X

ζ ∗ (ki − j, ki+1 , . . . , kr , k1 , . . . , ki−1 , j + 1) = k ζ (k + 1),

i=1 j=0

where the empty sum means zero. (Ohno and Okuda [873, p. 3030]) 49. Kamano [624] investigated the following multiple zeta function: ζn (s1 , . . . , sn ; a) =

X 05m1 <···<mn

1 (m1 + a)s1 · · · (mn + a)sn

(a)

a > 0; (m1 , . . . , mn ) ∈ Z ; (s1 , . . . , sn ) ∈ C , n

n




where Z denotes the set of integers. The special case n = 1 of ζn (s1 , . . . , sn ; a) in (a) reduces to the Hurwitz (or generalized) zeta function ζ (s, a). Also ζn (s1 , . . . , sn ; 1)

The Zeta and Related Functions

239

becomes the Euler-Zagier multiple zeta function denoted by ζn (s1 , . . . , sn ). Matsumoto [806] proved the analytic continuation of a more general class of multiple zeta functions, including (a) as a special case. Kamano [624] presented three kinds of limiting values of ζn (s1 , . . . , sn ; a) in (a) at nonpositive integers. Show that ζn0 (0 ; a) =

 n−1  (−1)n−1 Y 1 0(a) k+a− log √ (n − 1)! 2 2π k=1 ∂ ∂s ζn (s ; a), ζn (s ; a) := ζn (s, . . . , s ; a)

where ζn0 (s ; a) = stood to be nil.

(n ∈ N), in (a) and an empty sum is under(Kamano [624, Theorem 3])

50. Show that z γ (z) =

∞ X

(−1)k

k=2

Li(z) k

(|z| 5 1),

where γ (z) denotes the generalized-Euler-constant function, defined by γ (z) =

∞ X

zn−1



n=1

Z1 Z1 = 0 0

1 n+1 − log n n



(|z| 5 1)

1−x dxdy (1 − xyz)(− log xy)

(C \ [1, ∞)).

(Sondow and Hadjicostas [1053, Theorem 1]) 51. Let {αn }n∈N0 be a positive sequence, such that the following infinite series: ∞ X

e−αn t

n=0

converges for any t ∈ R+ . Then, for the generalized Hurwitz–Lerch Zeta function (ρ ,...,ρ ,σ ,...,σ )

8λ11,...,λpp;µ11,...,µqq (z, s, a), defined by 2.6(44), show that (ρ ,...,ρ ,σ ,...,σ )

8λ11,...,λpp;µ11,...,µqq (z, s, a) =

∞ Z∞

1 X 0(s)

  t s−1 e−(a−α0 +αn )t 1 − e−(αn+1 −αn )t

n=0 0

 · p 9q∗ 

(λ1 , ρ1 ), . . . , (λp , ρp ); (µ1 , σ1 ), . . . , (µq , σq );

 ze−t  dt


min{<(a), <(s)} > 0 ,

provided that each member exists, p 9q∗ being the Fox-Wright hypergeometric function defined in Problem 64 (Chapter 1). (Srivastava et al. [1107])

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Zeta and q-Zeta Functions and Associated Series and Integrals

52. For the generalized Hurwitz–Lerch Zeta function (ρ ,...,ρ ,σ ,...,σ )

8λ11,...,λpp;µ11,...,µqq (z, s, a), defined by 2.6(44), derive the following extension of the fractional derivative formulas, such as the one given by 2.6(32): o n (ρ ,...,ρ ,σ ,...,σ ) Dzν−τ zν−1 8λ11,...,λpp;µ11,...,µqq (zκ , s, a) =

0(ν) τ −1 (ρ1 ,...,ρp ,κ,σ1 ,...,σq ,κ) κ z 8λ1 ,...,λp ,ν;µ1 ,...,µq ,τ (z , s, a) 0(τ )


<(ν) > 0; κ > 0 .

(Srivastava et al. [1107]) 53. Show that each of the following integral representations holds true:

(ρ ,...,ρ ,σ ,...,σ ) 8λ11,...,λpp;µ11,...,µqq (z, s, a) =

1 0(s)

  Z∞ (λ1 , ρ1 ), . . . , (λp , ρp ); t s−1 e−at p 9q∗  ze−t  dt (µ , σ ), . . . , (µ , σ ); q q 1 1 0
min{<(a), <(s)} > 0

and q Q (ρ ,...,ρ ,σ ,...,σ )

8λ11,...,λpp;µ11,...,µqq (z, s, a) =

j=1 p Q

0 µj 0 λj







j=1

·

1 2πi

s Z 0(−ξ ) {0(ξ + a)} L

{0(ξ + a + 1)}s

p Q

j=1 q Q

0 λj + ρj ξ

0 µ j + σj ξ


(−z)ξ dξ


| arg(−z)| < π ,




j=1

where the path of integration L is a Mellin-Barnes type contour in the
√ complex ξ -plane, which starts at the point −i∞ and terminates at the point i∞ i := −1 with indentations, if necessary, in such a manner as to separate the poles of 0(−ξ ) from the
poles of 0 λj + ρj ξ ( j = 1, . . . , p). (Srivastava et al. [1107]) 54. For a given sequence {an }n∈N0 , let the operator 1 be defined by 10 an := an

and

1j an := 1j−1 an − 1j−1 an+1 =

j X k=0

(−1)k

  j an+k k

( j ∈ N).

Then derive the following analytic continuation formula for the Riemann Zeta function ζ (s) for all s ∈ C \ {1}: ζ (s) =

∞ X 1j 1−s 1 . 1 − 21−s 2j+1 j=0

The Zeta and Related Functions

241

Also, deduce the following special case, which was used in Hardy’s celebrated proof of the Riemann functional equation 2.3(11): 1 ζ (s) = 1 − 21−s

"

#  ∞ 1 1X 1 n−1 1 + (−1) − 2 2 ns (n + 1)s


<(s) > −1; s 6= 1 .

n=1

(Cf. Sondow [1048]) 55. For the extended Fermi-Dirac function 2v (s; x), defined in terms of the Weyl fractional s by integral operator Wx+

s 2v (s; x) := Wx+ [ϑ(t; v)] =

1 0(s)

Z∞ t s−1 ϑ(x + t; v)dt 0

Z∞ 1 = (t − x)s−1 ϑ(t; v)dt 0(s) x

  e−vt ϑ(t; v) := t ; <(s) > 0; x = 0; <(v) > −1 , e +1 show that 1 2v (µ + ν; x) = 0(µ)

Z∞ tµ−1 2v (ν; x + t)dt = 0

1 0(ν)

Z∞ tν−1 2v (µ; x + t)dt 0


min{<(µ), <(ν)} > 0; <(v) > −1 and 2v (s; x) =

∞ X

2v (s − n; 0)

n=0

(−x)n n!


s 6= n + 1 (n ∈ N0 ); x = 0; <(v) > −1 ,

provided that each member of these assertions exist. (See, for details, Srivastava et al. [1093]) 56. For the extended Bose-Einstein function 2v (s; x), defined in terms of the Weyl fractional s , by integral operator Wx+

s 9v (s; x) := Wx+ [ψ(t; v)] =

1 0(s)

Z∞ t s−1 ψ(x + t; v)dt 0

Z∞ 1 = (t − x)s−1 ψ(t; v)dt 0(s) x   e−vt ψ(t; v) := t ; <(s) > 1; x = 0; <(v) > −1 , e −1

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Zeta and q-Zeta Functions and Associated Series and Integrals

show that

9v (s; x) = q−s

q X

9 v+j−q (s; qx)


<(s) > 1; x = 0; <(v) > −1 .

q

j=1

(See, for details, Srivastava et al. [1093]) 57. For the extended Fermi-Dirac function 2v (s; x) and extended Bose-Einstein function 2v (s; x) (see Problems 55 and 56 above), derive each of the following identities: 22v (s; x) = 92v (s; x) − 21−s 9v (s, 2x)


<(s) > 1; x = 0; <(v) > −1 ,

i h 2v+1 (s, x) = 2−s 9 2v (s; 2x) − 9 v+1 (s; 2x)

<(s) > 1; x = 0; <(v) > −1

2v+1 (s; x) + 2v (s; x) = (v + 1)−s e−(v+1)x


<(s) > 0; x = 0; <(v) > −1 .

2




and

(See, for details, Srivastava et al. [1093]) (α) 58. For the Apostol-Genocchi polynomials Gn (x; λ) (λ ∈ C), defined by 1.8(58), derive each of the following explicit series representations:  Gn

p 2π iξ ;e q

 =

 q      2 · n! X n (2ξ + 2j − 1)p 2ξ + 2j − 1 exp − πi ζ n, (2qπ )n 2q 2 q j=1

+

     q X n (2j − 2ξ − 1)p 2j − 2ξ − 1 exp − + πi ζ n, 2q 2 q j=1


n ∈ N \ {1}; p ∈ Z; q ∈ N; ξ ∈ R , Gn(α)



p 2π iξ ;e q

 =

  n   X 2n 2 · k! n (α−1) 2π iξ (α−1) 2π iξ G e + G (e ) e2πiξ + 1 n−1 (2qπ)k k n−k k=2

X      q 2ξ + 2j − 1 k (2ξ + 2j − 1)p · ζ k, exp − πi 2q 2 q j=1

+

     q X k (2j − 2ξ − 1)p 2j − 2ξ − 1 exp − + πi ζ k, 2q 2 q j=1



n ∈ N \ {1}; p ∈ Z; q ∈ N; ξ ∈ R \ 3



   1 3 := k + : k ∈ Z ; α ∈ C 2

The Zeta and Related Functions

243

and Gn(α)



   n p 2πi n (−2)α (α−1)  2π iξ )  X k! n (α−1)  2π iξ  ;e = − 2πiξ Bn−1 −e − B −e q e +1 (2qπ )k k n−k k=2       q X k (2ξ + 2j − 1) p 2ξ + 2j − 1 exp − πi · ζ k,  2q 2 q j=1       q X 2j − 2ξ − 1 k (2j − 2ξ − 1) p + ζ k, exp − + πi  2q 2 q j=1      1 n ∈ N \ {1}; p ∈ Z; q ∈ N; ξ ∈ R \ 3 3 := k + : k ∈ Z ; α ∈ C 2

in terms of the Hurwitz (or generalized) Zeta function ζ (s, a). (See Luo and Srivastava [791, p. 5713, Theorem 4; p. 5715, Theorems 6 and 7])

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3 Series Involving Zeta Functions The main purpose of this chapter is to present a rather extensive collection of closedform sums of series involving the Zeta functions. Many of these summation formulas will find their applications in Chapter 5 (especially Section 5.3) in the evaluations of the determinants of the Laplacians for the n-dimensional sphere Sn with the standard metric. We begin this chapter by presenting an interesting historical introduction to the remarkably widely investigated subject of closed-form summation of series involving the Zeta functions. Among the various methods and techniques used in the vast literature on the subject, we give here reasonably detailed accounts of those using the binomial theorem, generating functions, multiple Gamma functions and hypergeometric identities. The last section (Section 3.6) deals with other methods, based (for example) on the Weierstrass canonical product form for the Gamma function and higher-order derivatives of the Gamma function.

3.1 Historical Introduction A rather classical (over two centuries old) theorem of Christian Goldbach (1690– 1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700–1782), was revived in 1986 by Shallit and Zikan [1022] as the following problem: X

(ω − 1)−1 = 1,

(1)

ω∈S

where S denotes the set of all nontrivial integer kth powers, that is, n o S := nk | n, k ∈ N \ {1} .

(2)

In terms of the Riemann Zeta function ζ (s), defined by 2.3(1), Goldbach’s theorem (1) assumes the elegant form (cf. Shallit and Zikan [1022, p. 403]): ∞ X

{ζ (k) − 1} = 1

k=2 Zeta and q-Zeta Functions and Associated Series and Integrals. DOI: 10.1016/B978-0-12-385218-2.00003-7 c 2012 Elsevier Inc. All rights reserved.

(3)

246

Zeta and q-Zeta Functions and Associated Series and Integrals

or, equivalently, ∞ X

F (ζ (k)) = 1,

(4)

k=2

where F (x) := x − [x] denotes the fractional part of x ∈ R. As a matter of fact, it is fairly straightforward to observe also that ∞ X k=2 ∞ X

1 (−1)k F (ζ (k)) = , 2 F (ζ (2k)) =

k=1

3 4

and

(5) ∞ X k=1

1 F (ζ (2k + 1)) = . 4

(6)

Another remarkable result involving the Riemann’s ζ -function is the following series representation for ζ (3): ( ) ∞ X π2 ζ (2k) ζ (3) = 1−4 7 (2k + 1)(2k + 2) 22k

(7)

k=1

or, equivalently,

ζ (3) = −

∞ ζ (2k) 4π 2 X , 7 (2k + 1)(2k + 2) 22k

(8)

k=0

since ζ (0) = − 12 (see Equation 2.3(10)). The series representation (7) is contained in a 1772 paper by Leonhard Euler (1707–1783) (see, e.g., Ayoub [81, pp. 1084–1085]). It was rediscovered by Ramaswami [966] and (more recently) by Ewell [436]. (See also Srivastava [1072, p. 7, Equation (2.23)]), where Euler’s result (7) was reproduced actually from the work of Ramaswami [966]). Numerous further series representations for ζ (3), which are analogous to (7) or (8), can be found in the works of Wilton [1233], Zhang and Williams [1250, 1251], Cvijovic´ and Klinowski [351], Chen and Srivastava [252], Srivastava [1082–1084, 1086], Srivastava and Tsumura [1111, 1112] and others (cf., e.g., Tsumura [1169, p. 384] and Ewell [438, p. 1004]); see also Berndt [122]). A considerably large variety of methods were used in the aforementioned works, as well as in the works of (among others) Jensen [607], Dinghas [385], Srivastava [1068, 1069, 1072, 1073], Klusch [673], Choi et al. [308], Choi and Srivastava [288, 289, 292, 294, 295], Choi and Seo [285] and Kanemitsu et al. [628], dealing with summation of series involving the Riemann ζ -function and its such extensions as the (Hurwitz’s) generalized Zeta function ζ (s, a), defined by 2.2(1).

Series Involving Zeta Functions

247

3.2 Use of the Binomial Theorem Landau’s formula (cf. Landau [729, p. 274, Eq. (3)]; see also Titchmarsh [1151, p. 33, Eq. (2.14.1)]): ∞

ζ (s) = 1 +

X (s)k 1 − {ζ (s + k) − 1} s−1 (k + 1)!

(1)

k=1

is capable of providing the analytic continuation of ζ (s) over the whole complex s-plane; here, (s)k denotes the Pochhammer symbol, defined by 1.1(5). Another formula, which can also be used in a similar way, is attributed to Ramaswami [966, p. 166] (see also Titchmarsh [1151, p. 33, Eq. (2.14.2)]): (1 − 21−s )ζ (s) =

∞ X (s)k ζ (s + k) . k! 2s+k

(2)

k=1

Motivated by these well-known results (1) and (2) in the theory of the Riemann zeta function ζ (s), Singh and Verma [1033] derived the following infinite series involving ζ (s): ∞

1 1X k · (s)k+1 1 + (−1)k−1 ζ (s + k + 1) (<(s) < 1) ζ (s) = + 2 s−1 2 (k + 2)!

(3)

k=1

and ∞

ζ (s) = 1 +

k · (s)k+1 1 s+3 1 X + (−1)k−1 {ζ (s + k + 1) − 1}. s+1 s−1 2 (k + 2)! 2

(4)

k=1

The proofs of (3) and (4) by Singh and Verma [1033, Sections 2 and 3] depend rather heavily on the integral representation (see Titchmarsh [1151, p. 14, Eq. (2.1.4)]): 1 1 ζ (s) = + −s 2 s−1

 Z∞ 1 dx x − [x] − (<(s) > −1). 2 xs+1

(5)

1

Srivastava [1069] gave relatively simple proofs of (3) and (4), without using the integral representation (5). He was, thus, led naturally to an interesting unification (and generalization) of (3) and (4), involving the Hurwitz (or generalized) Zeta function ζ (s, a) (see Section 2.2). The elementary techniques employed in Srivastava [1069] are shown to apply also to the derivation of numerous other results for ζ (s, a), including, for example, some useful analogues of (1) and (2).

248

Zeta and q-Zeta Functions and Associated Series and Integrals

The derivation of Srivastava’s unification (and generalization) of (3) and (4) is based simply upon the familiar binomial expansion (see Equation 1.5(22)): ∞ X (λ)k k=0

k!

zk = (1 − z)−λ

(|z| < 1).

(6)

Indeed, it follows readily from (6) and the definition 2.2(1) of the generalized Zeta function ζ (s, a) that (cf., e.g., Wilton [1233, p. 90, Eq. (1)]) ∞ X (λ)k k=0

k!

ζ (λ + k, a) tk = ζ (λ, a − t)

(|t| < |a|),

(7)

which holds true, by the principle of analytic continuation of ζ (s, a), for all values of λ 6= 1. Observe that (7) is the special case of 2.5(33) when x ∈ Z. Now, replace the summation index in (7) by k + 2, set λ = s − 1 and divide both sides of the resulting equation by t2 . We, thus, observe from (7) that ∞ X (s − 1)k+2 k=0

(k + 2)!

ζ (s + k + 1, a)tk = {ζ (s − 1, a − t) − ζ (s − 1, a)}t−2 − (s − 1)ζ (s, a)t−1

(8)

(0 < |t| < |a|).

Differentiating both sides of (8) with respect to t and noticing from the definition 1.1(5) and 2.2(18) that (s − 1)k+2 = (s − 1)(s)k+1

and

∂ {ζ (s − 1, a − t)} = (s − 1)ζ (s, a − t), (9) ∂t

we have ∞ X k · (s)k+1 k=1

(k + 2)!

ζ (s + k + 1, a)tk−1 = {ζ (s, a − t) + ζ (s, a)}t−2 (10)

2 − {ζ (s − 1, a − t) − ζ (s − 1, a)}t−3 s−1

(0 < |t| < |a|),

which can also be obtained as the special case of 2.5(34) when x ∈ Z. For t = −1, (10) readily yields the desired unification (and generalization) of (3) and (4) in the form (see Srivastava [1069, p. 49, Eq. (2.6)]): ζ (s, a) = a

−s



 ∞ 1 a 1X k · (s)k+1 + + (−1)k−1 ζ (s + k + 1, a), 2 s−1 2 (k + 2)! k=1

provided that the series converges.

(11)

Series Involving Zeta Functions

249

In the special case of (11) when a = 1, the series converges, if <(s) < 1, and we immediately obtain (3). Furthermore, in view of 2.3(2) and the special case of 2.3(9) when n = 1, the formula (11) with a = 2 is precisely the same as the known result (4). It may be of interest to remark here that alternative proofs of the well-known results (1) and (2), based upon the integral representation 2.3(30), were given by Menon [823]. As a matter of fact, both (1) and (2) can also be deduced fairly easily from (7). Replacing the summation index n in (7) by n + 1 and setting λ = s − 1, we have ∞ X k=0

(s)k ζ (s + k, a)tk+1 (k + 1)!

(12)

1 = {ζ (s − 1, a − t) − ζ (s − 1, a)} (|t| < |a|), s−1 where we have made use of the first identity in (9). By virtue of the definition 2.2(1), (12) with t = 1 assumes the form: ∞

ζ (s, a) =

(a − 1)1−s X (s)k − ζ (s + k, a), s−1 (k + 1)!

(13)

k=1

which, in view of 2.3(9) when n = 1, yields Landau’s formula (1) for a = 2. Conversely, (12) with t = 21 (and s replaced by s + 1) or (7) with t = 12 (and λ = s) similarly yields ζ (s, 2a − 1) − 21−s ζ (s, a) =

∞ X (s)k ζ (s + k, a) , k! 2s+k

(14)

k=1

which leads us immediately to Ramaswami’s formula (2) upon setting a = 1. For t = −1, (12) yields ∞

ζ (s, a) =

(s)k a1−s X + (−1)k−1 ζ (s + k, a), s−1 (k + 1)!

(15)

k=1

and (12) with t = − 12 (and s replaced by s + 1) or (7) with t = − 12 (and λ = s) gives ζ (s, 2a) − 21−s ζ (s, a) = −

∞ X (s)k ζ (s + k, a) (−1)k−1 . k! 2s+k

(16)

k=1

Formulas (15) with a = 2 and (16) with a = 1 provide interesting analogues of (1) and (2), respectively; in fact, this indicated analogue of (2) [that is, (16) with a = 1] was also given by Ramaswami [966, p. 166]. It is not difficult to deduce (11) as a natural consequence of (15). Numerous other consequences of the general results (7), (11) and (12) can be deduced by suitably specializing the parameter t in a manner detailed above.

250

Zeta and q-Zeta Functions and Associated Series and Integrals

In terms of the binomial coefficient 1.1(40), (6) and (7) with a = 2 are written in the following equivalent forms:  ∞  X λ+k−1 k=0

k

tk = (1 − t)−λ

(|t| < 1)

(17)

and  ∞  X λ+k−1 k=0

k

{ζ (λ + k) − 1}tk = ζ (λ, 2 − t)

(|t| < 2),

(18)

respectively. This last identity (18) is, in fact, equivalent to (cf. Ramanujan [964, p. 78, Eq. (15)]); Apostol [61, p. 240, Eq. (7)])  ∞  X λ+k−1 ζ (λ + k)tk = ζ (λ, 1 − t) k

(|t| < 1).

(19)

k=0

For fixed λ 6= 1, the series in (18) and (19) converge absolutely for |t| < 2 and |t| < 1, respectively. Thus, by the principle of analytic continuation, the formulas (18) and (19) are valid for all values of λ 6= 1. Formula (18) provides a unification (and generalization) of 3.1(4) and 3.1(5) and, indeed, also of a fairly large number of other summation formulas scattered in the literature. For example, in view of the relationships 2.3(1) and 2.2(4), (18) with t = 1 gives us (cf. Hansen [531, p. 356, Eq. (54.4.1)])  ∞  X λ+k−1 {ζ (λ + k) − 1} = 1, k

(20)

k=1

which generalizes 3.1(4), and a special case of (18) when t = −1 yields (cf. Hansen [531, p. 356, Eq. (54.4.2)])   ∞ X λ+k−1 (−1)k−1 {ζ (λ + k) − 1} = 2−λ , k

(21)

k=1

which generalizes 3.1(5). Several additional consequences of the general summation formulas (18) and (19) are worthy of note. First, replace the summation index k in (18) by k + 1 and set λ = s − 1, so that  ∞  X s+k−1 {ζ (s + k) − 1}tk+1 = ζ (s − 1, 2 − t) − ζ (s − 1) + 1 k+1

(|t| < 2),

k=0

(22)

Series Involving Zeta Functions

251

which, for t = 1, reduces immediately to the following alternative form of (20)  ∞  X s+k−1 k+1

k=0

{ζ (s + k) − 1} = 1.

(23)

Now, it follows from the definition 1.1(20) of the binomial coefficient that   s+k−1 (s − 1)(s)k (k ∈ N0 ). = (k + 1)! k+1

(24)

Thus, the formula (23) can easily be rewritten as Landau’s formula (1). For t = −1, (22) readily yields (see Srivastava [1073, p. 131, Eq. (2.4)]): ζ (s) = 1 +

∞

X 1 (s)k + (−1)k−1 {ζ (s + k) − 1}, s−1 s−1 (k + 1)! 2 1

(25)

k=1

which provides an interesting companion of Landau’s formula (1). Setting t = 21 in (22) and making use of 2.2(4) with a = 12 and n = 1, we obtain another series representation for ζ (s) (see Srivastava [1072, p. 5, Eq. (2.13)]): ∞

ζ (s) =

2s − 1 1 X (s)k + {ζ (s + k) − 1}. 2s − 2 2s − 2 k!2k

(26)

k=1

In their special cases when s = 2, (1) and (25) reduce simply to the summation formulas (4) and (5), respectively, whereas (26) similarly yields the elegant sum: ∞ X k−1 k=2

2k

{ζ (k) − 1} =

π2 − 1. 8

(27)

Next, we turn to the summation formula (19), which (for λ = s − 1 and with k replaced by k + 1) assumes the form:  ∞  X s+k−1 k=0

k+1

ζ (s + k)tk+1 = ζ (s − 1, 1 − t) − ζ (s − 1)

(|t| < 1).

(28)

In view of the identities in 2.3(1) and (24), a special case of (28) when t = 21 readily yields Ramaswami’s result (2). In case we add (19) to itself (with t replaced by −t), we obtain the summation formula (cf. Hansen [531, p. 357, Eq. (54.6.3)]):  ∞  X λ + 2k − 1 k=0

2k

1 ζ (λ + 2k)t2k = {ζ (λ − 1, 1 − t) + ζ (λ, 1 + t)} (|t| < 1), 2 (29)

252

Zeta and q-Zeta Functions and Associated Series and Integrals

whereas a similar subtraction yields  ∞  X λ + 2k 1 ζ (λ + 2k + 1)t2k+1 = {ζ (λ, 1 − t) − ζ (λ, 1 + t)} (|t| < 1). 2 2k + 1

(30)

k=0

Various interesting special cases of (29) and (30) are given in the literature. In particular, the special cases of (29) when t = 12 , t = 31 and t = 61 were considered by Ramaswami [966, p. 167, Eqs. (1), (3), and (4)], who also gave a special case of (30) when t = 12 (cf. Ramaswami [966, p. 167, Eq. (2)]), and by Apostol [61], who proved various generalizations of Ramaswami’s results. By assigning suitable numerical values to the variable s in some of the aforementioned special cases of (29) and (30), Ramaswami [966] also evaluated a number of special sums, including various formulas in Section 3.4 and 3.1(7). Wilton [1233, p. 92] showed that ∞ X (2k − 1)! ζ (2k) k=1

ζ (3) 1 11 + log π − 72 2π 2 12

(31)

1 1 ζ 0 (2) (2k)! ζ (2k + 1) 3 1 log 2 + 2 . = − log π − γ + 2k+1 (2k + 3)! 2 8 6 4 12 π

(32)

(2k + 3)! 22k

=

and ∞ X k=1

Many closed-form evaluations of series involving the Zeta function were investigated systematically by Srivastava [1072], who gave a rather detailed discussion about their derivations and relationships (if any) among them. As a matter of fact, most of the series identities considered by Srivastava [1072] will be included in our extensive list given in Section 3.4. We record, here, the following identity, involving the Bernoulli polynomials: n   Bn+1 (a + t) X n Bk+1 (a) n−k tn+1 = t + k k+1 n+1 n+1

(n ∈ N0 ),

(33)

k=0

which is, in fact, an immediate consequence of the known formula 1.7(13). It follows from 2.2(15) that the Laurent series expansion of ζ (s, a) at s = 1 is of the form ∞

ζ (s, a) =

X 1 − ψ(a) + cn (s − 1)n , s−1

(34)

n=1

where the coefficients cn are constants to be determined and the Psi (or Digamma) function ψ(z) is given in Section 1.3.

Series Involving Zeta Functions

253

Start with the following known identity for the Hurwitz-Lerch Zeta function 8(z, s, a), defined by 2.5(1) (see 2.5(33) and 2.6(9)): ∞ X (s)k k=0

k!

8(z, s + k, a) tk = 8(z, s, a − t)

(|t| < |a|; s 6= 1),

(35)

where (λ)n denotes the Pochhammer symbol, defined by 1.1(5). The general series identity, which is to be proven in this section, is contained in Theorem 3.1 below (see [297]). Theorem 3.1 For every nonnegative integer n, ∞ X

(−1)n 0 tn+k [8 (z, −n, a − t) − 80 (z, −n, a)] = (k)n+1 n! k=2   n X  (−1)n+k n  + (Hn − Hn−k ) 8(z, k − n, a) − 80 (z, k − n, a) tk n! k 8(z, k, a)

(36)

k=1

+ [Hn L1 (z, a) − L2 (z, a)]

tn+1 (n + 1)!

(|t| < |a|; |z| < 1; n ∈ N0 ),

where Hn denotes the harmonic numbers, defined by Hn :=

n X 1 j=1

j

,

(37)

it being understood (as elsewhere in this paper) that an empty sum is nil, 80 (z, s, a) =

∂ 8(z, s, a), ∂s

and L1 (z, a) and L2 (z, a) are defined by L1 (z, a) := lim {(s + n) 8(z, s + n + 1, a)}

(38)

L2 (z, a) := lim

(39)

s→−n

and s→−n



8(z, s + n + 1, a) + (s + n) 80 (z, s + n + 1, a) .

Before proceeding to prove Theorem 3.1, we show that the assertion (36) of Theorem 3.1 is equivalent to that of Theorem 3.2 below, by making use of the elementary identity: n X Anj 1 1 = := , (k)n+1 k(k + 1) · · · (k + n) k+j j=0

(40)

254

Zeta and q-Zeta Functions and Associated Series and Integrals

where Anj

  (−1)j n = (0 5 j 5 n; j, n ∈ N0 ); n! j

the combinatorial identities:   n X 1 (−1)j n = j+1 j n+1 j=0 n X j=1

(n ∈ N0 ),

  Hn (−1)j+1 n Hj = j+1 j n+1

(41)

(n ∈ N0 ),

(42)

and n X j=n−k+1

    (−1)j+1 n n = (−1)n+k (Hn − Hn−k ) j − (n − k) j k

(0 5 k 5 n; n, k ∈ N0 ), (43)

the special case k = n of which is recorded in [505, p. 5, Entry 0.155]; the familiar result:    n X n j j (−1) = 0 (0 5 k 5 n − 1; k ∈ N0 ; n ∈ N), (44) k j j=k

and some rather simple manipulations, using such elementary series identities as the ones involved in j−1 n X X

Aj,k =

j=0 k=0

n−1 X n X

(45)

Aj,k

k=0 j=k+1

and j n X X j=0 k=0

Aj,k =

n X n X

Aj,k ,

(46)

k=0 j=k

where {Aj,k } ( j, k ∈ N0 ) is a suitably bounded double sequence. Theorem 3.2 For every nonnegative integer n, n   n−1 ∞ X X 8(z, −k, a) n−k 8(z, k, a) k+n X n t = 80 (z, −k, a − t) tn−k − t k+n k n−k k=2

k=0

tn+1 − [Hn L1 (z, a) + L2 (z, a)] − 80 (z, −n, a) n+1

k=0

(47)

(|t| < |a|; |z| < 1; n ∈ N0 ),

where L1 (z, a) and L2 (z, a) are given as in (38) and (39), respectively.

Series Involving Zeta Functions

255

It follows easily from the definitions 1.1(19) and 1.3(1) that d {(z)n } = (z)n [ψ(z + n) − ψ(z)]. dz

(48)

Proof of Theorem 3.1 Upon transposing the first n + 2 terms from k = 0 to k = n + 1 in (35) to the right-hand side, if we divide both sides of the resulting equation by s + n, we get ∞ X

(s)n (s + n + 1)k−n−1 8(z, s + k, a)

k=n+2

gn (z, s, t, a) tk = k! s+n

(49)

(|t| < |a|; s 6= 1; n ∈ N0 ), where, for convenience, gn (z, s, t, a) := 8(z, s, a − t) −

n+1 X (s)k k=0

k!

8(z, s + k, a) tk .

(50)

Now we shall show that lim gn (z, s, t, a) = 0.

(51)

s→−n

Indeed, we observe that 8(z, −n, a − t) =

∞ X

(k + a − t)n zk

k=0

=

 ∞ X n   X n k=0

=

=



n  X j=0 n X j=0

j=0

j

(k + a)n−j (−t)j

 

zk



 ∞ X n zk (−t)j j (k + a)−n+j k=0

(−n)j 8(z, −n + j, a) tj , j!

from which (51) follows easily. Thus, by l’Hoˆ pital’s rule, we have lim

s→−n

gn (z, s, t, a) ∂ = lim {gn (z, s, t, a)}. s→−n ∂s s+n

(52)

256

Zeta and q-Zeta Functions and Associated Series and Integrals

Next, by appealing to (50) and (48), we find that ∂ {gn (z, s, t, a)} = 80 (s, a − t) − 80 (s, a) ∂s n X tn+1 tk , − h(z, s, a, k) − h(z, s, a, n + 1) k! (n + 1)!

(53)

k=1

where, for convenience, h(z, s, a, k) := (s)k [{ψ(s + k) − ψ(s)} 8(z, s + k, a) + 80 (z, s + k, a)].

(54)

In view of 1.3(7), (54) yields   lim h(z, s, a, k) = −(−n)k (Hn − Hn−k ) 8(z, k − n, a) − 80 (z, k − n, a)

s→−n

(k = 1, . . . , n)

(55)

and lim h(z, s, a, n + 1)    n X (s) n  (s + n) 8(z, s + n + 1, a) + (s)n+1 80 (z, s + n + 1, a), = lim  s→−n s+j

s→−n

j=0

(56) which, upon writing n−1 n X X (s)n (s)n (s)n = + , s+j s+j s+n j=0

j=0

reduces to the following form: lim h(z, s, a, n + 1) = (−n)n [−Hn L1 (z, a) + L2 (z, a)],

s→−n

(57)

where L1 (z, a) and L2 (z, a) are given by (38) and (39), respectively. Conversely, upon taking the limit on the left-side of (49) as s → −n, we have lim

s→−n

∞ X

(s)n (s + n + 1)k−n−1 8(z, s + k, a)

k=n+2

= (−n)n

∞ X k=2

tk+n 8(z, k, a) . (k)n+1

tk k! (58)

Finally, upon taking the limit on both sides of (49) as s → −n and considering (52), (53), (54), (57) and (58), we are led immediately to the desired series identity (36). This evidently completes our proof of Theorem 3.1.

Series Involving Zeta Functions

257

Applications of Theorems 3.1 and 3.2 We first note that 8(z, s, a) is an entire function of s when |z| < 1, converges for <(s) > 0 when |z| = 1 and z 6= 1 and reduces to ζ (s, a) when z = 1. So, when |z| 5 1 and z 6= 1, we have L1 (z, a) = 0

and L2 (z, a) = 8(z, 1, a) = a−1 2 F1 (1, a ; 1 + a ; z).

(59)

It follows from 2.5(26) that L1 (1, a) = lim (s + n) ζ (s + n + 1, a) = 1,

(60)

s→−n

since ζ (s, a) has a simple pole at s = 1 with residue 1. From (34), we get L2 (1, a) = −ψ(a).

(61)

Upon setting z = 1 in (36) and (47) and considering 2.5(26), (60) and (61), we have certain families of series associated with the generalized (or Hurwitz) Zeta function ζ (s, a): Corollary 3.3 For every nonnegative integer n, ∞ X ζ (k, a) n+k (−1)n 0 t = [ζ (−n, a − t) − ζ 0 (−n, a)] (k)n+1 n! k=2   n X  (−1)n+k n  + (Hn − Hn−k ) ζ (k − n, a) − ζ 0 (k − n, a) tk n! k

(62)

k=1

+ [Hn + ψ(a)]

tn+1 (n + 1)!

(|t| < |a|; n ∈ N0 ),

where Hn denotes the harmonic numbers defined by (37), it being understood that ζ 0 (s, a) =

∂ ζ (s, a). ∂s

Corollary 3.4 For every nonnegative integer n, ∞ n   n−1 X X ζ (k, a) k+n X n 0 ζ (−k, a) n−k t = ζ (−k, a − t) tn−k − t k+n k n−k k=2

k=0

k=0

tn+1 + [ψ(a) − Hn ] − ζ 0 (−n, a) n+1

(63)

(|t| < |a|; n ∈ N0 ),

where Hn denotes the harmonic numbers defined by (37), it being understood that ζ 0 (s, a) =

∂ ζ (s, a). ∂s

258

Zeta and q-Zeta Functions and Associated Series and Integrals

Infinite sums of the type occurring in (62) can also be evaluated, in a markedly different manner, in terms of such higher transcendental functions as the multiple Gamma functions (see, for details, [628, p. 10, Theorem 3.1]). The series identities (62) and (63) can include a number of formulas involving various classes of earlier closed-form evaluations of series associated with the Zeta function as its special cases. For example, we can get the following (easily derivable) consequence of the summation formula (63): ∞ X ζ (2k, a) k=1

k+n

t

2k+2n

2n   h i X 2n = ζ 0 (−k, a − t) + (−1)k ζ 0 (−k, a + t) t2 n−k k k=0

(64)

n−1 X ζ (−2`, a) 2n−2` t − 2 ζ 0 (−2n, a) − n−`

(n ∈ N0 ; |t| < |a|);

`=0

∞ X ζ (2k, a) 2k+2n+1 t 2k + 2n + 1 k=1

 2n+1  i 1 X 2n + 1 h 0 = ζ (−k, a − t) + (−1)k ζ 0 (−k, a + t) t2n+1−k 2 k

(65)

k=0

−

n X ζ (−2`, a) 2n+1−2` t 2n + 1 − 2`

(n ∈ N0 ; |t| < |a|);

`=0

∞ X ζ (2k + 1, a) 2k+2n+1 t 2k + 2n + 1 k=1

2n   i 1 X 2n h 0 = ζ (−k, a − t) − (−1)k ζ 0 (−k, a + t) t2n−k k 2 k=0

n X ζ (1 − 2`, a) 2n−2`+1 t2n+1 [ψ(2n + 1) − ψ(a) + γ ] − t − 2n − 2` + 1 2n + 1

(66)

`=1

(n ∈ N0 ; |t| < |a|); ∞ X ζ (2k + 1, a) 2k+2n+2 t k+n+1 k=1

=

2n+1 X k=0 n X

−

`=1

 i 2n + 1 h 0 ζ (−k, a − t) − (−1)k ζ 0 (−k, a + t) t2n+1−k k

ζ (1 − 2`, a) 2n+2−2` t2n+2 [ψ(2n + 2) − ψ(a) + γ ] t − n−`+1 n+1

− 2 ζ 0 (−2n − 1, a)

(n ∈ N0 ; |t| < |a|).

(67)

Series Involving Zeta Functions

259

Setting a = 1 and t = −1 in (63) and using various identities given in the previous chapters, for example, 2.2(4) and 1.3(7), we obtain ∞ X (−1)k k=2

k+n

ζ (k) =

n−1 X `=0

  n−1 X n 0 (−1)` (−1) ζ (−`) − ζ (−`) ` rn − ` `

`=0

1 + (γ + Hn ) n+1

(68)

(n ∈ N0 ).

Likewise, setting a = 1 and t = 1/2 in (63), we get ∞ X k=2

   n−1     X n n 2` ζ (−`) + 1 − 2` ζ 0 (−`) log 2 − n−` ` ` `=0 `=0 (69)  

n−1

X ζ (k) = k (k + n) 2

γ + Hn + ζ (−n) log 2 + 1 − 2n+1 ζ 0 (−n) 2(n + 1)

−

(n ∈ N0 ).

The special cases of (64) and (65) when a = 1 and t = 1/2 are written here: ∞ X k=1

ζ (2k) 1 − 22n+1 1 − log 2 + (−1)n (2n)! ζ (2n + 1) = 2k 2n (2π)2n (k + n) 2

+

n−1 X k=1

∞ X k=1

   2n (2k)!  2k (−1) 1 − 2 ζ (2k + 1) 2k (2π)2k k

(70) (n ∈ N).

1 1 ζ (2k) = − log 2 2(2n − 1) 2 (2k + 2n − 1) 22k

  n−1  1X (2k)!  k 2n − 1 2k + (−1) 1 − 2 ζ (2k + 1) 2k 2 (2π)2k

(71) (n ∈ N),

k=1

the special case n = 1 of which yields 3.4(519). Setting a = 1 and t = 1/2 in (66), we get ∞ X

 n n  X X ζ (2k + 1) 22`−1 B2` 2n B2` = − log 2 2k `(2n − 2` + 1) 2` − 1 ` (2k + 2n + 1) 2 `=1 `=1 k=1   n   X 2n +2 1 − 22`−1 ζ 0 (1 − 2`) 2` − 1 `=1

  2n X γ + H2n ` 2n − + log 2 (−1) 2n + 1 ` `=0

(n ∈ N0 ),

(72)

260

Zeta and q-Zeta Functions and Associated Series and Integrals

where we have assumed that   2n X 2n (−1)` := 1 (n = 0). ` `=0

Of course, it is true that 2n X

(−1)`

`=0

  2n = 0 (n ∈ N). `

(73)

Setting a = 1 and t = 1/2 in (67), we obtain  ∞ n  X X ζ (2k + 1) 2n + 1 B2`+2 γ + H2n+1 = −2 log 2 − n+1 2` + 1 ` + 1 (k + n + 1) 22k `=0

k=1

+

n X

22`−1

B2` + 4

n−1  X

1 − 22`+1

`(n − ` + 1) `=0   2n+2 0 +4 1−2 ζ (−2n − 1) (n ∈ N0 ). `=1

 2n + 1 2` + 1

ζ 0 (−2` − 1)

(74)

If we set n = 1 in (74), we get ∞ X ζ (2k + 1) 2 γ 29 =− − − log 2 + 12 log A + 30 log C. 3 2 30 (k + 2) · 22k

(75)

k=1

Upon setting a = 1 in (65) and then t = 21 in the resulting equation, if we use the various identities for ζ (s) and ζ (s, a) given in Chapter 2, in particular, together with the known identity 2.3(22), we obtain (see [256, p. 262, Eq. (14)]) ∞ X k=1

ζ (2k) 1 1 = − log 2 2(2` − 1) 2 (2k + 2` − 1) 22k

  `−1  1X (2k)!  k 2` − 1 2k + (−1) 1 − 2 ζ (2k + 1) 2 2k (2π)2k

(76) (` ∈ N),

k=1

whose special case ` = 1 is also recorded in 3.4(519). As a consequence, the following integral can be evaluated as a finite series involving the Riemann Zeta function (see [256, Eq. (18)]): π

Z2

θ 2` cot θ dθ =

0

+

`−1 X k=1

 π 2`  22`+1 − 1 log 2 + (−1)` (2`)! ζ (2` + 1) 2 (2π)2`

#    (2k)!  k+1 2` 2k (−1) 1−2 ζ (2k + 1) 2k (2π)2k

(` ∈ N).

(77)

Series Involving Zeta Functions

261

We note that the integral in (77) was evaluated earlier as an infinite series involving the Riemann Zeta function (see [505, p. 428, Entry 3.748.2]; also see [256]). We deduce another interesting identity by suitably combining the special cases of (62) when a = 1 and a = 2. By applying 2.2(4), 1.3(4) and 1.3(7), we, thus, find that ∞ X (−1)n+1 tn+k = (1 − t)n log(1 − t) (k)n+1 n! k=1   n X (−1)n+k n + (Hn − Hn−k ) tk n! k

(78) (|t| < 1; n ∈ N0 ),

k=1

which, in the special case when n = 2, is a known result recorded (for example) by Hansen [531, p. 37, Entry (5.7.40); p. 74, Entry (5.16.26)]. We shall deal with other interesting applications of Corollary 3.3 in Section 3.4.

3.3 Use of Generating Functions Adamchik and Srivastava [11] proposed and developed a novel method of evaluation of the sums of series involving Zeta functions by using generating functions. The key ingredients in their approach happen to include the familiar integral representations 2.2(21) and 2.3(30). This method has already been implemented in Mathematica (Version 3.0). Consider the sum: (a) =

∞ X

f (k) ζ (k + 1, a)

(<(a) > 0),

(1)

k=1

where the sequence { f (n)}∞ n=1 is assumed to possess a generating function: F(t) =

∞ X

f (k)

k=1

tk k!

(2)

and f (n) = O

  1 n

(n → ∞).

(3)

Upon replacing the Hurwitz (or generalized) Zeta function in (1) by its integral representation given by 2.2(21) with s = k + 1, if we invert the order of summation and integration, we obtain (a) =

Z∞ F(t) 0

e−(a−1)t dt, et − 1

where we have also made use of the generating function (2).

(4)

262

Zeta and q-Zeta Functions and Associated Series and Integrals

Thus, the problem of summation of series of the type (1) has been reduced formally to that of integration in (4). Although the integral in (4) appears to be fairly involved for symbolic integration, it may be possible to reduce it to 2.2(21) or 2.3(30) (or another known integral), especially when F(t) is a power, exponential, trigonometric or hyperbolic function. The first example (Proposition 3.5 below) would illustrate how this technique actually works (see Adamchik and Srivastava [11, p. 135, Proposition 1]): Proposition 3.5 Let n be a positive integer. Then,   n−1 ∞  Y  X (−1)k {ζ (nk) − 1} = log  0 2 − (−1)(2j+1)/n . k

(5)

j=0

k=1

Remark 1 In Proposition 3.5, as also in Equations (7), (8) and (10) below, it is tacitly assumed that −1 = elog(−1) , where log z denotes the principal branch of the logarithmic function in the complex z-plane for which −π < arg(z) 5 π

(z 6= 0).

Proof. Denote, for convenience, the left-hand side of the summation formula (5) by 2(n). Then, in view of the special case of 2.3(9) when n = 1, we can apply the integral representation 2.2(21) with s = nk and a = 2. Upon inverting the order of summation and integration, which can be justified by the absolute convergence of the series and the integral involved, we, thus, find that 2(n) = n

Z∞ 0

∞ X (−tn )k dt . tet (et − 1) 0(nk + 1)

(6)

k=1

By recognizing the series in (6) as a trigonometric function of order n (see, for details, Erde´ lyi et al. [422, Section 18.2]) or, alternatively, by using an easily derivable special case of a known result in Hansen [531, p. 207, Entry (10.49.1)], we have ∞ X k=1

n   (−tn )k 1X = −1 + exp t(−1)(2j+1)/n . 0(nk + 1) n

(7)

j=1

Substituting from (7) into (6) and inverting the order of summation and integration once again, we obtain 2(n) = lim

r→1−

Z∞

 −n

0

n

X dt + r t t t e (e − 1)

Z∞

j=1 0

 
 exp − 1 − (−1)(2j+1)/n t dt , tr (et − 1) (8)

Series Involving Zeta Functions

263

where the parameter r < 1 is inserted with a view to providing convergence of the integrals at their lower terminal t = 0. Finally, we evaluate each integral in (8) by means of 2.2(21) and proceed to the limit as r → 1−. Indeed, by making use of the known behavior of ζ (1 − r, a) near r = 1 for fixed a (cf., e.g., Erde´ lyi et al. [421, p. 26]):   0(a) 1 (9) ζ (1 − r, a) ∼ − a + (1 − r) log √ (r → 1−; a fixed), 2 2π we arrive at the right-hand side of the assertion (5). In precisely the same manner, we can prove a mild generalization of Proposition 3.5, which we state here as (see Adamchik and Srivastava [11, p. 136, Proposition 2])  Proposition 3.6 Let n be a positive integer. Then, ∞ X (−1)k ζ (nk, a) = −n log 0(a) k k=1   n−1  Y  + log  0 a − (−1)(2j+1)/n 

(10)
<(a) = 1 .

j=0

Next we prove (see Adamchik and Srivastava [11, p. 136, Proposition 3]) Proposition 3.7 Let n be a positive integer. Then, in terms of the Bernoulli numbers Bn , defined by 1.7(2), and the Stirling numbers S(n, k), defined by 1.6(14), ∞ X

(−1)k {ζ (k) − 1} kn = −1 +

k=2

−

n X

1 − 2n+1 Bn+1 n+1 (11) (−1) k! ζ (k + 1)S(n + 1, k + 1). k

k=1

Proof. Making use of the integral representation 2.2(21) with a = 2, if we invert the order of summation and integration and then evaluate the resulting integral and sum, we find that 3(n) :=

∞ X

(−1)k {ζ (k) − 1} kn

k=2



d = lim r r→1 dr

n

(12) {r [ψ(r + 2) − ψ(2)]} .

Since   n X d r {r f (r)} = r S(n + 1, k + 1) f (k) (r) rk+1 dr k=0

(13)

264

Zeta and q-Zeta Functions and Associated Series and Integrals

and (cf. Equation 1.2(53)) n o ψ (k) (3) = (−1)k k! 1 + 2−k−1 − ζ (k + 1)

(k ∈ N),

(14)

we find from (12) that 3(n) = −1 −

n X

(−1)k k! ζ (k + 1) S(n + 1, k + 1)

k=1

+

n X

(15)

(−1) k! 2 k

−k−1

S(n + 1, k + 1).

k=0

To complete the proof of Proposition 3.7, we, thus, need to show that n X

(−1)k k! 2−k−1 S(n + 1, k + 1) =

k=0

1 − 2n+1 Bn+1 n+1

(16)

in terms of the Bernoulli numbers Bn , defined by 1.7(2). In fact, it is known that (Hansen [531, p. 351, Entry (52.2.36)]) n X

(−1)k k! 2−k S(n, k) =

k=1

 2  1 − 2n+1 Bn+1 , n+1

(17)

which, in view of the recurrence relation 1.6(18) with k replaced by k + 1, yields the desired identity (16). Similarly, we can prove (see Adamchik and Srivastava [11, p. 137, Proposition 4]) 

Proposition 3.8 Let n be a positive integer. Then, ∞ X k=2

{ζ (k) − 1} kn = 1 +

n X

k! ζ (k + 1) S(n + 1, k + 1).

(18)

k=1

The following list provides further summation formulas, involving series of Zeta functions, which can be derived by applying the foregoing technique (see Adamchik and Srivastava [11, pp. 138–139]): ∞ X k2 3 γ π2 1 {ζ (k) − 1} = − + − log(2π); k+1 2 2 6 2

(19)

∞ X k2 9 1 {ζ (2k + 1) − 1} = − γ + log 2 − ζ (3); k+1 16 2

(20)

k=2

k=2

Series Involving Zeta Functions ∞ X

{ζ (4k) − 1} =

k=1

265

7 π − coth π ; 8 4

(21)

 √   √  sin π 2 + sinh π 2 π  √   √ ; (−1)k {ζ (4k) − 1} = 1 + √ 2 2 cos π 2 − cosh π 2 k=1

∞ X

∞ X

{ζ (4k) − 1} z4k =

k=1

3z4 − 1 πz {cot(πz) + coth(πz)} − 4 2(z4 − 1)

(|z| < 2),

(22)

(23)

which contains both (21) and (22) as limiting cases; ∞ X k=1

  1 1 {ζ (2k) − 1} sin k = − cot 2 2

π sin + 2

  1 2

 ∞  X p+k k

k=1

        sin 2π cos 21 − cos 12 sinh 2πsin 12       ; cos 2π cos 12 − cosh 2π sin 12

ζ (p + k + 1, a) zk =

o (−1)p n (p) ψ (a) − ψ (p) (a − z) p! (p ∈ N; <(a) > 0; |z| < |a|);

(24)

(25)

  ∞ X k 3 = 8G ζ k + 1, 4 2k

(26)

  ∞ X k 5 ζ k + 1, = 8(1 − G), 4 2k

(27)

k=2

and

k=2

where G denotes Catalan’s constant, defined by 1.3(16). Remark 2 Since the right-hand side of the relationship 1.3(53) (with n! replaced by 0(n + 1)) is well-defined for n ∈ C \ {−1}, the summation formula (25) may be put in a slightly more general form (cf. Wilton [1233]; see also Srivastava [1072, p. 137, Equation (6.6)]):  ∞  X s+k−1 k=0

k

ζ (s + k, a) zk = ζ (s, a − z)


s ∈ C \ {1}; a 6= Z− 0 ; |z| < |a| ,

which, in view of 1.1(20), is equivalent to 3.2(7) for λ = s and t = z.

(28)

266

Zeta and q-Zeta Functions and Associated Series and Integrals

The foregoing method has been implemented in Mathematica (Version 3.0), as we remarked at the beginning of this section.

Series Involving Polygamma Functions (s)

The generalized harmonic numbers Hn , defined by (cf. Graham et al. [507]) Hn(s)

n X 1 := ks

(n ∈ N, s ∈ C),

(29)

k=1

which, by virtue of 2.3(9), can be written in the form: Hn(s) = ζ (s) − ζ (s, n + 1)

(<(s) > 1; n ∈ N).

(30)

In view of the relationships 1.2(53) and (30), the foregoing techniques can be applied also to series involving Polygamma functions and generalized harmonic numbers. We first state (see Adamchik and Srivastava [11, p. 139, Proposition 5]) Proposition 3.9 Let p be a positive integer. Then, in terms of the function 8(z, s, a), defined by 2.5(1), ∞ X

ψ (p) (a + k) zk =

k=1

(−1)p+1 p! ψ (p) (a + 1) + 1−z ap+1

(−1)p p! z2 + 8(z, p + 1, a + 1) 1−z

|z| < 1;

(31)


a 6= Z− 0 \ {0} .

Proof. Denoting, for convenience, the left-hand side of the summation formula (31) by 4(z), it is not difficult to find from 1.3(53) and 2.2(21) that 4(z) = (−1)

p+1

Z∞ 0

Z1 =− 0

tp e−(a−1)t et − 1

τ a−1 (log τ )p dτ (1 − τ )(1 − zτ )

∞ X

! k −kt

z e

dt

k=1

(<(a) > 0).

Upon evaluating this last integral and waiving the restriction on the parameter a by appealing to the principle of analytic continuation, we complete the proof of Proposition 3.9. Next, we turn to a family of linear harmonic sums: Sp,q :=

∞ (p) X Hn , nq n=1

(32)

Series Involving Zeta Functions

267

which were discussed extensively by Flajolet and Salvy [454]. By applying (30), 2.2(21) and 2.3(30), it is not difficult to prove (see Adamchik and Srivastava [11, p. 140, Proposition 6])  Proposition 3.10 Let Sp,q be defined by (32). Then, (−1)p Sp,q = ζ (p)ζ (q) + (p − 1)!

Z1

(−1)q Sp,q = ζ (p + q) − (q − 1)!

Z1

(log t)p−1 Liq (t)

dt 1−t

(33)

(log t)q−1 Lip (t)

dt . 1−t

(34)

0

and

0

Remark 3 In view of the following symmetry relation in Flajolet and Salvy [454]: Sp,q + Sq,p = ζ (p)ζ (q) + ζ (p + q),

(35)

the integral representations (33) and (34) are essentially the same. Although, in general, the integrals occurring in (33) and (34) cannot be evaluated in closed forms, many interesting particular cases of linear harmonic sums would follow from Proposition 3.10.

Series Involving Polylogarithm Functions We shall derive, among other results, an integral representation for the Khintchine constant K0 , which arises in the measure theory of continued fractions. Every positive irrational number µ can, indeed, be written uniquely as a simple continued fraction as follows: µ = a0 +

a1 a3 an ··· ··· , a2 + a4 + an+1 +

(36)

that is, with a0 a non-negative integer and with all other aj (j ∈ N) positive integers. The Gauss-Kuz’min distribution (cf., e.g., Khintchine [647]) predicts that the density of occurrence of some chosen positive integer k in the continued fraction (36) of a random real number is given by  Prob{an = k} = −log2

 1 . 1− (k + 1)2

(37)

And, making use of the Gauss-Kuz’min distribution involving (37), Khintchine [647] showed that, for almost all irrational numbers, the limiting geometric mean of the

268

Zeta and q-Zeta Functions and Associated Series and Integrals

positive integer elements aj (j ∈ N) of the relevant continued fraction exists and equals K0 :=

∞  Y 1+ k=1

1 k(k + 2)

log2 k (38)

  ∞  Y 1 log 1+ k(k+2) . = k 2 k=1

An interesting explicit representation of the Khintchine constant K0 in terms of Polylogarithm functions was proven recently by Bailey et al. [85, p. 422]:     ∞ 1 1X 4 log(K0 ) log 2 = (log 2)2 + Li2 − + (−1)n Li2 2 . 2 2 n

(39)

n=2

If we set L(n) :=

∞ X

(−1) Li2 n



n=2

 4 , n2

(40)

replace the Dilogarithm function by its series representation given by 2.4(3), change the order of summation and evaluate the inner sum, we shall obtain L(n) = 2

∞ ∞ X ζ (2k) X 4k {ζ − (2k) − 1} . k2 k2 k=1

(41)

k=1

It seems very unlikely that the sums occurring in (41) can be evaluated in terms of well-known functions. Nevertheless, by noting that 1

Z ∞ k ∞ X t dt X tk ζ (2k) = ζ (2k) t k k2 k=1

(42)

k=1

0

and evaluating the inner sum by the method illustrated in the preceding sections, we find that ∞ k  √   √ X t ζ (2k) = log π t csc π t . k2

(43)

k=1

Combining (41), (42) and (43), we have L(n) :=

∞ X n=2

(−1) Li2 n



4 n2

Z1

 =

0

√ √
! π t cot π t dt , log t 1 − 4t

(44)

Series Involving Zeta Functions

269

which leads us immediately to the following integral representation for the Khintchine constant K0 (see Adamchik and Srivastava [11, p. 141, Eq. (4.10)]): π 2 (log 2)2 + + log(K0 ) log 2 = 12 2

Zπ

log (t| cot t|)

dt . t

(45)

0

Other sums involving the Polylogarithm function, which were also evaluated by Adamchik and Srivastava [11], are given below.  ∞    X 1 k − Lik z2 = 1 − z−1 arctanh z; 2

(46)

k=1

∞  k X 1 k=1

2

Since arctanh z=

    1 1+z Lik z2 = z log . 2 1−z

  1 2

(47)

= log 3, both (46) and (47) can be expressed in terms of log 3 when

1 2.

3.4 Use of Multiple Gamma Functions We provide a rather extensive list of evaluations of series involving the Zeta functions, by making use of the Gamma and multiple Gamma functions. Although our list in this section includes some of the series identities considered in previous sections or earlier works, we choose to present, here, all of the identities evaluated by the use of multiple Gamma functions for the sake of completeness.

Evaluation by Using the Gamma Function Upon setting n = 0 in 3.2(35) and applying 2.2(17), we obtain the familiar result (cf. Whittaker and Watson [1225, p. 276], Hansen [531, p. 358, Entry (54.11.1)], and Srivastava [1072, p. 18]): ∞ X k=2

ζ (k, a)

tk = log 0(a − t) − log 0(a) + tψ(a) k

(|t| < |a|),

(1)

which, upon replacing t by −t, yields ∞ X tk (−1)k ζ (k, a) = log 0(a + t) − log 0(a) − tψ(a) k k=2

(|t| < |a|).

(2)

270

Zeta and q-Zeta Functions and Associated Series and Integrals

By adding and subtracting, we find from (1) and (2) that ∞ X

ζ (2k, a)

k=1 ∞ X

t2k = log 0(a + t) + log 0(a − t) − 2 log 0(a) k

ζ (2k + 1, a)

k=1

(|t| < |a|);

1 t2k+1 = {log 0(a − t) − log 0(a + t)} + tψ(a) 2k + 1 2

(3)

(|t| < |a|). (4)

Differentiating both sides of (1), (2), (3) and (4) with respect to t, we have ∞ X

ζ (k, a)tk−1 = −ψ(a − t) + ψ(a)

(|t| < |a|);

(5)

k=2 ∞ X (−1)k ζ (k, a)tk−1 = ψ(a + t) − ψ(a)

(|t| < |a|);

(6)

k=2 ∞ X

ζ (2k, a)t2k−1 =

k=1 ∞ X k=1

1 {ψ(a + t) − ψ(a − t)} 2

(|t| < |a|);

1 ζ (2k + 1, a)t2k = − {ψ(a + t) + ψ(a − t)} + ψ(a) 2

(7)

(|t| < |a|).

(8)

Setting a = 1 in (1) through (8) and using 1.3(4) and 2.3(2), we obtain ∞ X k=2

ζ (k)

tk = log 0(1 − t) − γ t k

(|t| < 1);

∞ X tk (−1)k ζ (k) = log 0(1 + t) + γ t k

(9)

(|t| < 1);

(10)

k=2

∞ X k=1 ∞ X k=1 ∞ X k=2

ζ (2k)

t2k = log 0(1 + t) + log 0(1 − t) k

ζ (2k + 1)

(|t| < 1);

t2k+1 1 = {log 0(1 − t) − log 0(1 + t)} − γ t 2k + 1 2

ζ (k)tk−1 = −ψ(1 − t) − γ

(|t| < 1);

(11)

(|t| < 1);

(12)

(13)

Series Involving Zeta Functions

271

∞ X (−1)k ζ (k)tk−1 = ψ(1 + t) + γ

(|t| < 1);

(14)

k=2 ∞ X

ζ (2k)t2k−1 =

k=1 ∞ X k=1

1 {ψ(1 + t) − ψ(1 − t)} 2

(|t| < 1);

1 ζ (2k + 1)t2k = − {ψ(1 + t) + ψ(1 − t)} − γ 2

(15)

(|t| < 1).

(16)

Equations (11) and (15) can, in view of 1.1(12) and 1.3(9), be written in their respective equivalent forms: ∞ X k=1 ∞ X

  t2k πt ζ (2k) = log k sin π t ζ (2k)t2k−1 = −

k=1

(|t| < 1);

π 1 cot(πt) + 2 2t

(17)

(|t| < 1).

(18)

If we use some elementary trigonometric identities, we obtain the following trigonometric evaluations: π 



7π = sin sin 8 8



q √ 1 = 2 − 2; 2 (19)

q √ 1 = 2 + 2; 2 √     p π  4π 3π 5− 5 = sin sin = cos = ; √ 5 5 10 2 2 p √     π  2π 3π 5+ 5 ; sin = sin = cos = √ 5 5 10 2 2     √ π  4π 3π 5+1 cos = −cos = sin = ; 5 5 10 4      π  √5 − 1 2π 3π cos = ; = −cos = sin 5 5 10 4 



3π sin 8

cot

π  5



5π = sin 8



4π = −cot 5

π 



r

 =

2√ 1+ 5; 5

2π cot 5



√ 5 + 2 5;







(20)

(21)



3π = −cot 5

r

 =

1−

2√ 5; 5 (22)



9π cot = −cot 10 10

q

 =

3π cot 10



7π = −cot 10

q

 =

√ 5 − 2 5.

272

Zeta and q-Zeta Functions and Associated Series and Integrals

Taking the limit as t → 1 on both sides of (10), we have the well-known result: ∞ X ζ (k) = γ. (−1)k k

(23)

k=2

For suitable special values of the argument t, we can deduce the following series identities from (9) through (18), by making use of such evaluations as (19) through (22), as well as the ψ-function values listed in Section 1.3. ∞ X ζ (k) 1 γ = log π − ; k 2 2 k·2 k=2 ∞ X

ζ (k) 1 γ = + log π − log 2; 2 2 k · 2k k=2 ∞ π  X ζ (2k) ; = log 2 k · 22k (−1)k

(24) (25) (26)

k=1

∞ X ζ (2k + 1) = log 2 − γ ; (2k + 1)22k k=1   ∞ X ζ (2k) 2π = log √ ; k · 32k 3 3 k=1     ∞ 2k X ζ (2k) 2 4π = log √ ; k 3 3 3 k=1   ∞ X ζ (2k) π = log √ ; k · 24k 2 2 k=1     ∞ X ζ (2k) 3 2k 3π = log √ ; k 4 2 2 k=1 ! √ ∞ X ζ (2k) 2 2π p = log √ ; k · 52k 5 5− 5 k=1 ! √   ∞ X ζ (2k) 2 2k 4 2π p = log √ ; k 5 5 5+ 5 k=1 ! √   ∞ X ζ (2k) 3 2k 6 2π p = log √ ; k 5 5 5+ 5 k=1 ! √   ∞ X ζ (2k) 4 2k 8 2π p = log √ ; k 5 5 5+ 5 k=1    √ ∞  5 + 1 π X ζ (2k) = log ;   10 k · 102k k=1

(27) (28) (29) (30) (31)

(32)

(33)

(34)

(35)

(36)

Series Involving Zeta Functions

273

   √  2k ∞  3 5 − 1 π X ζ (2k) 3 ; = log   k 10 10 k=1    √  2k ∞  7 5 − 1 π X ζ (2k) 7 ; = log   k 10 10 k=1    √   ∞ 9 5+1 π  X ζ (2k) 9 2k = log ;   k 10 10

(37)

(38)

(39)

k=1

∞ X ζ (2k) k=1

k · 62k

= log

π  ; 3

  ∞ X ζ (2k) 5 2k

(40) 

 5π = log ; k 6 3 k=1 ! ∞ X ζ (2k) π = log p √ ; k · 26k 4 2− 2 k=1 !   ∞ X ζ (2k) 3 2k 3π = log p √ ; k 8 4 2 + 2 k=1 !   ∞ X 5π ζ (2k) 5 2k = log p √ ; k 8 4 2+ 2 k=1 q     ∞ X √ 7π ζ (2k) 7 2k = log 4+2 2 ; k 8 8 k=1   √ ∞ π 3 + 1 X ζ (2k) ; = log  √ k · 122k 6 2 k=1   √   ∞ 3−1 5π X ζ (2k) 5 2k ; = log  √ k 12 6 2 k=1   √   ∞ 7π 3−1 X ζ (2k) 7 2k ; = log  √ k 12 6 2 k=1 √    2k ∞ 11π 3 + 1 X ζ (2k) 11 ; = log  √ k 12 6 2

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

k=1

∞ X ζ (k) (−1)k k = 1 − log 2; 2 k=2

(50)

274

Zeta and q-Zeta Functions and Associated Series and Integrals ∞ X ζ (k) k=2 ∞ X k=1 ∞ X k=1 ∞ X

2k

= log 2;

ζ (2k) = 1; 22k−1

(52)

ζ (2k + 1) = 2 log 2 − 1; 22k

(53)

(−1)k

k=2 ∞ X k=2 ∞ X k=1 ∞ X

1 ζ (k) 1 √ = 1 − π 3 − log 3; k 18 2 3

(55)

ζ (2k) 1 1 √ = − π 3; 2 18 32k

(56)

(−1)k

k=2 ∞ X k=2 ∞ X k=1 ∞ X

ζ (k) 1 3 = 1 − π − log 2; 2k 8 4 2

(57)

(58)

(59)

(60)

(61)

(62)

ζ (k) 3 1 = log 2 − π; 4 8 22k

(63)

ζ (2k) 1 1 = − π; 2 8 24k

(64)

ζ (2k + 1) = 3 log 2 − 2; 24k k=1  k ∞ X 3 3 9 k (−1) ζ (k) = 1 + π − log 2; 4 8 4 k=2

(54)

ζ (k) 1 1 √ = log 3 − π 3; k 2 18 3

ζ (2k + 1) 3 = (log 3 − 1); 2 32k k=1  k ∞ X 1 √ 2 (−1)k ζ (k) = 1 + π 3 − log 3; 3 9 k=2   ∞ X 2 k 1 √ ζ (k) = π 3 + log 3; 3 9 k=2  2k ∞ X 2 1 1 √ ζ (2k) = + π 3; 3 2 9 k=1  2k ∞ X 3 2 3 ζ (2k + 1) = log 3 − ; 3 2 4 k=1 ∞ X

(51)

(65)

(66)

Series Involving Zeta Functions ∞ X

 k 3 3 9 = π + log 2; 4 8 4 k=2   ∞ X 3 2k 1 3 = + π; ζ (2k) 4 2 8 k=1   ∞ X 3 2k 2 ζ (2k + 1) = 3 log 2 − ; 4 3 k=1 r √ √ ∞ X 2√ 1 5 π 1+ 5 k ζ (k) 1+ 5 − log 5 − (−1) k = 1 − log ; 10 5 4 10 2 5 k=2 r √ √ ∞ X 5 ζ (k) 2√ 1 1+ 5 π 1+ 5 + log 5 + log ; =− 10 5 4 10 2 5k k=2 r ∞ X ζ (2k) 1 π 2√ = − 1 + 5; 2 10 5 52k k=1 √ √ ∞ X ζ (2k + 1) 5 5 5 1+ 5 = − + log 5 + log ; 2 4 2 2 52k k=1 r √ √  k ∞ X π 2√ 1 5 1+ 5 2 k (−1) ζ (k) = 1− 1− 5 − log 5 + log ; 5 5 5 2 5 2 k=2 r √ √  k ∞ X 2√ 1 5 π 1+ 5 2 1− 5 + log 5 − ζ (k) =− log ; 5 5 5 2 5 2 k=2 r  2k ∞ X 2 1 π 2√ ζ (2k) = − 1− 5; 5 2 5 5 k=1 √ √  2k ∞ X 1+ 5 2 5 5 5 log ; ζ (2k + 1) = − + log 5 − 5 4 4 2 2 k=1 r √ √  k ∞ X 3 3π 2√ 3 3 5 1+ 5 k (−1) ζ (k) log ; = 1+ 1− 5 − log 5 + 5 10 5 4 10 2 k=2 r √ √  k ∞ X 3 3π 2√ 3 3 5 1+ 5 ζ (k) = 1− 5 + log 5 − log ; 5 10 5 4 10 2 k=2 r  2k ∞ X 3 1 3π 2√ ζ (2k) = + 1− 5; 5 2 10 5 k=1 √ √  2k ∞ X 3 5 5 5 1+ 5 ζ (2k + 1) = − + log 5 − log ; 5 6 4 2 2 k=1 r √ √  k ∞ X 4 2π 2√ 2 5 1+ 5 k (−1) ζ (k) = 1+ 1+ 5 − log 5 − log ; 5 5 5 5 2 k=2

ζ (k)

275

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

(75)

(76)

(77)

(78)

(79)

(80)

(81)

(82)

276

Zeta and q-Zeta Functions and Associated Series and Integrals

r √ √  k 4 2π 2√ 2 5 1+ 5 ζ (k) = 1+ 5 + log 5 + log ; 5 5 5 5 2 k=2 r  2k ∞ X 1 2π 2√ 4 = + 1+ 5; ζ (2k) 5 2 5 5 k=1 √ √  2k ∞ X 4 5 5 5 1+ 5 ζ (2k + 1) = − + log 5 + log ; 5 8 4 2 2 ∞ X

k=1 ∞ X

(−1)k

k=2 ∞ X k=2 ∞ X k=1 ∞ X

1 ζ (k) 1 √ 1 = 1 − π 3 − log 3 − log 2; k 12 4 3 6

k=1 ∞ X

(85)

(86)

(87)

ζ (2k) 1 1 √ = − π 3; 2 12 62k

(88)

 ζ (2k) 1 π √ = − 2 + 1 ; 2 16 26k

√  √  ζ (2k + 1) 2 = −4 + 4 log 2 − log 3 − 2 2 ; 2 26k k=1 √  k ∞  3  X √  3 3π √ 3 2 k (−1) ζ (k) =1− 2 − 1 − log 2 − log 3 − 2 2 ; 8 16 2 16 k=2

(84)

ζ (k) 1 1 1 √ = − π 3 + log 3 + log 2; k 12 4 3 6

ζ (2k + 1) 3 = −3 + log 3 + 2 log 2; 2 62k k=1  k ∞ X 5 √ 5 5 5 (−1)k ζ (k) = 1 + π 3 − log 3 − log 2; 6 12 4 3 k=2   ∞ X 5 5 √ 5 5 k ζ (k) = π 3 + log 3 + log 2; 6 12 4 3 k=2  2k ∞ X 5 1 5 √ ζ (2k) = + π 3; 6 2 12 k=1  2k ∞ X 5 3 3 ζ (2k + 1) = − + log 3 + 2 log 2; 6 5 2 k=1 √ ∞   1 X √  π √ 2 k ζ (k) (−1) 3k = 1 − log 3 − 2 2 ; 2 + 1 − log 2 + 16 2 16 2 k=2 √ ∞  1  X ζ (k) √  π √ 2 = − 2 + 1 + log 2 − log 3 − 2 2 ; 16 2 16 23k k=2 ∞ X

(83)

(89)

(90)

(91)

(92)

(93)

(94)

(95)

(96)

(97)

(98)

Series Involving Zeta Functions

277

√  k  3  √  3 3π √ 3 2 ζ (k) =− 2 − 1 + log 2 + log 3 − 2 2 ; (99) 8 16 2 16 k=2  2k ∞  X 1 3π √ 3 = − 2−1 ; (100) ζ (2k) 8 2 16 k=1 √  2k ∞  X √  3 4 2 ζ (2k + 1) = − + 4 log 2 + log 3 − 2 2 ; (101) 8 3 2 k=1 √  k ∞  5  X √  5π √ 5 2 5 k 2 − 1 − log 2 − (−1) ζ (k) = 1+ log 3 − 2 2 ; 8 16 2 16

∞ X

k=2

(102) √  k     √ 5π √ 5 2 5 5 = log 3 − 2 2 ; ζ (k) 2 − 1 + log 2 + (103) 8 16 2 16 k=2  2k ∞  X 1 5π √ 5 = + 2−1 ; (104) ζ (2k) 8 2 16 k=1 √  2k ∞  X √  4 2 5 ζ (2k + 1) = − + 4 log 2 + log 3 − 2 2 ; (105) 8 5 2 k=1 √  k ∞  7  X √  7 7π √ 7 2 k (−1) ζ (k) =1 + 2 + 1 − log 2 + log 3 − 2 2 ; 8 16 2 16 ∞ X

k=2

(106) √  k     √ 7π √ 7 7 7 2 = ζ (k) 2 + 1 + log 2 − log 3 − 2 2 ; (107) 8 16 2 16 k=2  2k ∞  X 7 1 7π √ ζ (2k) = + 2+1 ; (108) 8 2 16 k=1 √  2k ∞  X √  7 2 4 ζ (2k + 1) log 3 − 2 2 ; (109) = − + 4 log 2 − 8 7 2 k=1 √ q ∞  X √ √  1 5 π 1 k ζ (k) 5 + 2 5 − log 2 − log 5 − (−1) = 1− log 2 + 5 ; k 20 5 8 20 10 ∞ X

k=2

(110) √ q  √ √  ζ (k) π 1 1 5 = − 5 + 2 5 + log 2 + log 5 + log 2 + 5 ; 20 5 8 20 10k k=2 q ∞ X √ ζ (2k) 1 π = − 5 + 2 5; 2k 2 20 10 k=1 √ ∞  X ζ (2k + 1) √  5 5 = −5 + 2 log 2 + log 5 + log 2 + 5 ; 4 2 102k

∞ X

k=1

(111)

(112)

(113)

278

Zeta and q-Zeta Functions and Associated Series and Integrals

q  k ∞ X √ 3 3π 3 (−1)k ζ (k) = 1− 5 − 2 5 − log 2 10 20 5 k=2 √  √  3 5 3 log 2 + 5 ; (114) − log 5 + 8 20 √ q   ∞  X √ √  3 k 3 3π 3 3 5 ζ (k) 5 − 2 5 + log 2 + log 5 − =− log 2 + 5 ; 10 20 5 8 20 k=2

(115) q √ 3 1 3π ζ (2k) = − 5 − 2 5; (116) 10 2 20 k=1 √  2k ∞  X √  5 5 5 3 = − + 2 log 2 + log 5 − log 2 + 5 ; ζ (2k + 1) (117) 10 3 4 2 k=1 q  k ∞ X √ 7 7π 7 k (−1) ζ (k) = 1+ 5 − 2 5 − log 2 10 20 5 k=2 √  √  7 5 7 log 2 + 5 ; − log 5 + (118) 8 20 √ q  k ∞  X √ √  7π 7 7 7 7 5 = ζ (k) 5 − 2 5 + log 2 + log 5 − log 2 + 5 ; 10 20 5 8 20 ∞ X



2k

k=2

(119) q √ 1 7π 7 = + ζ (2k) 5 − 2 5; (120) 10 2 20 k=1 √  2k ∞  X √  7 5 5 5 ζ (2k + 1) log 2 + 5 ; (121) = − + 2 log 2 + log 5 − 10 7 4 2 k=1 q  k ∞ X √ 9 9π 9 k (−1) ζ (k) = 1+ 5 + 2 5 − log 2 10 20 5 k=2 √  √  9 9 5 (122) − log 5 − log 2 + 5 ; 8 20 √ q  k ∞  X √ √  9 9 9 9 5 9π ζ (k) 5 + 2 5 + log 2 + log 5 + log 2 + 5 ; = 10 20 5 8 20 ∞ X



2k

k=2

(123) ∞ X

2k

q √ 9 1 9π ζ (2k) = + 5 + 2 5; 10 2 20 k=1 √  2k ∞  X √  9 5 5 5 ζ (2k + 1) = − + 2 log 2 + log 5 + log 2 + 5 ; 10 9 4 2 k=1



(124)

(125)

Series Involving Zeta Functions

279

√ ∞  X √  √  1 π  3 1 k ζ (k) (−1) = 1 − 2 + 3 + log 7 − 4 3 − log 3 + log 2; k 24 24 8 4 12 k=2

(126) ∞ X k=2 ∞ X k=1 ∞ X

√  √  √  1 π  3 1 ζ (k) = − 2 + 3 − log 7 − 4 3 + log 3 + log 2; k 24 24 8 4 12

(127)

√  π  ζ (2k) 1 = − 2 + 3 ; 2 24 122k

(128)

 √  3 ζ (2k + 1) 1√ = −6 − 3 log 7 − 4 3 + log 3 + 3 log 2; 2 2 122k k=1  k ∞ X 5 (−1)k ζ (k) 12 k=2 √  √  5 3 √  5 5π  5 = 1− 2− 3 − log 7 − 4 3 − log 3 − log 2; 24 24 8 4  k ∞ X 5 ζ (k) 12 k=2 √  √  5 3 √  5 5 5π  2− 3 + log 7 − 4 3 + log 3 + log 2; =− 24 24 8 4  2k ∞   X √ 5 1 5π ζ (2k) = − 2− 3 ; 12 2 24 k=1 √  2k ∞  X √  3 6 5 3 =− + log 7 − 4 3 + log 3 + 3 log 2; ζ (2k + 1) 12 5 2 2 k=1   ∞ X 7 k (−1)k ζ (k) 12 k=2 √  √  7 3 √  7 7 7π  2− 3 − log 7 − 4 3 − log 3 − log 2; = 1+ 24 24 8 4  k ∞ X 7 ζ (k) 12 k=2 √  √  7 3 √  7 7π  7 = 2− 3 + log 7 − 4 3 + log 3 + log 2; 24 24 8 4  2k ∞   X √ 7 1 7π ζ (2k) = + 2− 3 ; 12 2 24 k=1 √   ∞  X √  3 7 2k 6 3 ζ (2k + 1) =− + log 7 − 4 3 + log 3 + 3 log 2; 12 7 2 2 k=1

(129)

(130)

(131) (132)

(133)

(134)

(135)

(136)

(137)

280

Zeta and q-Zeta Functions and Associated Series and Integrals

 k ∞ X 11 (−1)k ζ (k) 12 k=2

√  √  11 3 √  11 11 11π  2+ 3 + log 7 − 4 3 − log 3 − log 2; = 1+ 24 24 8 4  k ∞ X 11 ζ (k) 12 k=2 √  √  11 3 √  11 11 11π  2+ 3 − log 7 − 4 3 + log 3 + log 2; = 24 24 8 4  2k ∞ X √  1 11π  11 = + 2+ 3 ; ζ (2k) 12 2 24 k=1 √  2k ∞  X √  3 6 11 3 =− − log 7 − 4 3 + log 3 + 3 log 2. ζ (2k + 1) 12 11 2 2

(138)

(139)

(140)

(141)

k=1

In view of the various identities given in previous sections and the special case of 2.3(9) when n = 1, by setting a = 2 in (1) through (8), we obtain the following additional series identities: ∞ X

{ζ (k) − 1}

k=2 ∞ X

tk = log 0(2 − t) + (1 − γ )t k

{ζ (2k) − 1}

k=1 ∞ X

(|t| < 2);

(143)

t2k = log 0(2 + t) + log 0(2 − t) k

(|t| < 2);

(144)

{ζ (2k + 1) − 1}

k=1

= ∞ X

t2k+1 2k + 1

1 {log 0(2 − t) − log 0(2 + t)} + (1 − γ )t 2

{ζ (k) − 1} tk−1 = −ψ(2 − t) + 1 − γ

k=2 ∞ X

k=1 ∞ X k=1

{ζ (2k) − 1} t2k−1 =

(145) (|t| < 2);

(|t| < 2);

(−1)k (ζ (k) − 1)tk−1 = ψ(2 + t) + γ − 1

k=2 ∞ X

(142)

tk = log 0(2 + t) + (γ − 1)t k

(−1)k {ζ (k) − 1}

k=2 ∞ X

(|t| < 2);

(146)

(|t| < 2);

1 {ψ(2 + t) − ψ(2 − t)} 2

(147)

(|t| < 2);

1 {ζ (2k + 1) − 1} t2k = − {ψ(2 + t) + ψ(2 − t)} + 1 − γ 2

(148) (|t| < 2);

(149)

Series Involving Zeta Functions ∞ X k=1 ∞ X

{ζ (2k) − 1}

281

  t2k π t(1 − t2 ) = log k sin π t

{ζ (2k) − 1} t2k−1 = −

k=1

(|t| < 2);

3t2 − 1 π cot(πt) + 2 2t(t2 − 1)

(|t| < 2).

(150)

(151)

Similarly, as above, setting various suitable arguments in (142) through (151), we readily obtain the following evaluations: for instance, by taking the limit in (143) as t → 2, we obtain ∞ X 2k (−1)k {ζ (k) − 1} = 2γ − 2 + log 6; k k=2  √  ∞ X ζ (k) − 1 γ 1 3 π (−1)k = − + log ; k 2 2 4 k·2 k=2 √  ∞ X ζ (k) − 1 1 γ π ; = − + log k 2 2 2 k·2 k=2   ∞ X ζ (2k) − 1 3π = log ; 8 k · 22k k=1   ∞ X ζ (2k + 1) − 1 2 = 1 − γ + log ; 3 (2k + 1)22k k=1 ∞ X

(−1)k

k=2 ∞ X k=2 ∞ X k=1 ∞ X

ζ (k) − 1 = γ − 1 + log 2; k

(153)

(154)

(155)

(156)

(157)

ζ (k) − 1 = 1−γ; k

(158)

ζ (2k) − 1 = log 2; k

(159)

ζ (2k + 1) − 1 1 = 1 − γ − log 2; 2k + 1 2 k=1    √  ∞ k X 3γ 3 15 π k ζ (k) − 1 3 (−1) = − + log ; k 2 2 2 8 k=2   ∞ X ζ (k) − 1 3 k 3 3γ 1 = − + log π ; k 2 2 2 2 k=2     ∞ X ζ (2k) − 1 3 2k 15π = log ; k 2 8 k=1

(152)

(160)

(161)

(162)

(163)

282

Zeta and q-Zeta Functions and Associated Series and Integrals

  ∞ X ζ (2k + 1) − 1 3 2k k=1 ∞ X

2k + 1

2 

= 1−γ +

  1 8 log ; 3 15

 16π ζ (2k) − 1 = log √ ; k · 32k 27 3 k=1     ∞ X 20π ζ (2k) − 1 2 2k = log √ ; k 3 27 3 k=1     ∞ X ζ (2k) − 1 4 2k 56π = log √ ; k 3 27 3 k=1     ∞ X ζ (2k) − 1 5 2k 160π = log √ ; k 3 27 3 k=1   ∞ X ζ (2k) − 1 15π = log √ ; k · 24k 32 2 k=1     ∞ X 21π ζ (2k) − 1 3 2k = log √ ; k 4 32 2 k=1     ∞ X ζ (2k) − 1 5 2k 45π = log √ ; k 4 32 2 k=1     ∞ X ζ (2k) − 1 7 2k 231π = log √ ; k 4 32 2 k=1 ! √ ∞ X ζ (2k) − 1 48 2 π p = log √ ; k · 52k 125 5 − 5 k=1 ! √   ∞ X ζ (2k) − 1 2 2k 84 2 π p = log √ ; k 5 125 5 + 5 k=1 ! √   ∞ X ζ (2k) − 1 3 2k 96 2 π p = log √ ; k 5 125 5 + 5 k=1 ! √   ∞ 2k X ζ (2k) − 1 4 72 2 π p = log √ ; k 5 125 5 − 5 k=1 ! √   ∞ X ζ (2k) − 1 6 2k 132 2 π p = log √ ; k 5 125 5 − 5 k=1 ! √   ∞ X ζ (2k) − 1 7 2k 336 2 π p = log √ ; k 5 125 5 + 5 k=1

(164)

(165)

(166)

(167)

(168)

(169)

(170)

(171)

(172)

(173)

(174)

(175)

(176)

(177)

(178)

Series Involving Zeta Functions

283

! √   ∞ X ζ (2k) − 1 8 2k 624 2 π p = log √ ; k 5 125 5 + 5 k=1 ! √   ∞ X ζ (2k) − 1 9 2k 1008 2 π p = log √ ; k 5 125 5 − 5 k=1    √ ∞  5 + 1 99 π X ζ (2k) − 1 = log ;   1000 k · 102k k=1 √      ∞  273 5−1 π  X ζ (2k) − 1 3 2k = log ;   k 10 1000 k=1 √     2k ∞  357 5 − 1 π X ζ (2k) − 1 7 = log ;   k 10 1000 k=1 √      ∞  171 5+1 π  X ζ (2k) − 1 9 2k = log ;   k 10 1000 k=1   √   2k ∞  5 + 1 π 231 X ζ (2k) − 1 11 = log ;   k 10 1000 k=1 √     2k ∞  897 5 − 1 π X ζ (2k) − 1 13 = log ;   k 10 1000 k=1   √    ∞  3213 5−1 π  X ζ (2k) − 1 17 2k = log ;   k 10 1000 k=1 √     2k ∞  4959 5 + 1 π X ζ (2k) − 1 19 = log ;   k 10 1000

(179)

(180)

(181)

(182)

(183)

(184)

(185)

(186)

(187)

(188)

k=1

∞ X ζ (2k) − 1



 35π = log ; 108 k · 62k k=1     ∞ X ζ (2k) − 1 5 2k 55π = log ; k 6 108 k=1     ∞ X 91π ζ (2k) − 1 7 2k = log ; k 6 108 k=1     ∞ X ζ (2k) − 1 11 2k 935π = log ; k 6 108 k=1

(189)

(190)

(191)

(192)

284

Zeta and q-Zeta Functions and Associated Series and Integrals ∞ X ζ (2k) − 1

! 63π p = log √ ; k · 26k 256 2 − 2 k=1 !   ∞ X ζ (2k) − 1 3 2k 165π p = log √ ; k 8 256 2 + 2 k=1 !   ∞ X ζ (2k) − 1 5 2k 195π p = log √ ; k 8 256 2 + 2 k=1 !   ∞ X 105π ζ (2k) − 1 7 2k p = log √ ; k 8 256 2 − 2 k=1 !   ∞ 2k X ζ (2k) − 1 9 153π p = log √ ; k 8 256 2 − 2 k=1 !   ∞ X ζ (2k) − 1 11 2k 627π p = log √ ; k 8 256 2 + 2 k=1 !   ∞ X 1365π ζ (2k) − 1 13 2k p = log √ ; k 8 256 2 + 2 k=1 !   ∞ X 2415π ζ (2k) − 1 15 2k p = log √ ; k 8 256 2 − 2 k=1 ! √ ∞ X ζ (2k) − 1 143π( 3 + 1) ; = log √ 2k k · 12 864 2 k=1 ! √   ∞ X ζ (2k) − 1 5 2k 595π( 3 − 1) = log ; √ k 12 864 2 k=1 ! √   ∞ X ζ (2k) − 1 7 2k 665π( 3 − 1) = log ; √ k 12 864 2 k=1 ! √   ∞ X ζ (2k) − 1 11 2k 253π( 3 + 1) ; = log √ k 12 864 2 k=1 ! √   ∞ X ζ (2k) − 1 13 2k 325π( 3 + 1) = log ; √ k 12 864 2 k=1 ! √   ∞ X ζ (2k) − 1 17 2k 2465π( 3 − 1) = log ; √ k 12 864 2 k=1 ! √   ∞ X ζ (2k) − 1 19 2k 4123π( 3 − 1) = log ; √ k 12 864 2 k=1

(193)

(194)

(195)

(196)

(197)

(198)

(199)

(200)

(201)

(202)

(203)

(204)

(205)

(206)

(207)

Series Involving Zeta Functions

  ∞ X ζ (2k) − 1 23 2k k=1 ∞ X

12

k

(−1)k

k=2

2k

1 = log 2 − ; 2

∞ X ζ (2k) − 1 k=1 ∞ X

! √ 8855π( 3 + 1) = log ; √ 864 2

ζ (k) − 1 5 = − log 2; 6 2k

∞ X ζ (k) − 1 k=2

285

22k

1 = ; 6

(209)

(210)

(211)

ζ (2k + 1) − 1 4 = − + 2 log 2; 2k 3 2

(212)

1 (−1)k {ζ (k) − 1} = ; 2

(213)

{ζ (k) − 1} = 1;

(214)

3 {ζ (2k) − 1} = ; 4

(215)

k=1 ∞ X

k=2 ∞ X

k=2 ∞ X k=1 ∞ X

1 {ζ (2k + 1) − 1} = ; 4 k=1   ∞ X 3 k 31 = − 3 log 2; (−1)k {ζ (k) − 1} 2 10 k=2  k ∞ X 3 3 {ζ (k) − 1} = + 3 log 2; 2 2 k=2   ∞ X 3 2k 23 {ζ (2k) − 1} = ; 2 10 k=1   ∞ X 3 2k 8 {ζ (2k + 1) − 1} = − + 2 log 2; 2 15 k=1 ∞ X

(−1)k

k=2 ∞ X k=2

(208)

ζ (k) − 1 11 π√ 1 3 − log 3; = − k 12 18 2 3

ζ (k) − 1 1 π√ 1 =− − 3 + log 3; k 6 18 2 3

(216)

(217)

(218)

(219)

(220)

(221)

(222)

286

Zeta and q-Zeta Functions and Associated Series and Integrals ∞ X ζ (2k) − 1 k=1 ∞ X

32k

=

3 1 √ − π 3; 8 18

13 3 ζ (2k + 1) − 1 = − + log 3; 2k 8 2 3 k=1   ∞ X 2 k 11 1 √ (−1)k (ζ (k) − 1) = + π 3 − log 3; 3 15 9 k=2  k ∞ X 4 1 √ 2 {ζ (k) − 1} = − + π 3 + log 3; 3 3 9 k=2   ∞ X 2 2k 3 1 √ {ζ (2k) − 1} = − + π 3; 3 10 9 k=1   ∞ X 31 3 2 2k {ζ (2k + 1) − 1} = − + log 3; 3 20 2 k=1  k ∞ X 89 2 √ 4 = (−1)k {ζ (k) − 1} − π 3 − 2 log 3; 3 21 9 k=2  k ∞ X 4 2 √ 4 {ζ (k) − 1} = − π 3 + 2 log 3; 3 3 9 k=2   ∞ X 4 2k 39 2 √ {ζ (2k) − 1} = − π 3; 3 14 9 k=1  2k ∞ X 4 61 3 {ζ (2k + 1) − 1} = − + log 3; 3 56 2 k=1  k ∞ X 5 √ 5 59 5 + π 3 − log 3; (−1)k {ζ (k) − 1} = 3 24 18 2 k=2   ∞ X 5 k 5 5 √ 5 {ζ (k) − 1} = + π 3 + log 3; 3 3 18 2 k=2   ∞ X 5 2k 33 5 √ {ζ (2k) − 1} = + π 3; 3 16 18 k=1  2k ∞ X 5 19 3 {ζ (2k + 1) − 1} = − + log 3; 3 80 2 k=1 ∞ X

(−1)k

k=2 ∞ X k=2

ζ (k) − 1 19 1 3 = − π − log 2; 2k 20 8 4 2

ζ (k) − 1 1 1 3 = − − π + log 2; 12 8 4 22k

(223)

(224)

(225)

(226)

(227)

(228)

(229)

(230)

(231)

(232)

(233)

(234)

(235)

(236)

(237)

(238)

Series Involving Zeta Functions ∞ X ζ (2k) − 1 k=1 ∞ X

24k

=

13 1 − π; 30 8

31 ζ (2k + 1) − 1 = − + 3 log 2; 4k 15 2 k=1   ∞ X 3 k 19 3 9 k (−1) {ζ (k) − 1} = + π − log 2; 4 28 8 4 k=2  k ∞ X 9 3 9 3 {ζ (k) − 1} = − + π + log 2; 4 4 8 4 k=2   ∞ X 3 2k 11 3 {ζ (2k) − 1} = − + π; 4 14 8 k=1   ∞ X 41 3 2k {ζ (2k + 1) − 1} = − + 3 log 2; 4 21 k=1  k ∞ X 191 5 5 15 = (−1)k {ζ (k) − 1} − π− log 2; 4 36 8 4 k=2  k ∞ X 5 5 15 5 {ζ (k) − 1} = − π+ log 2; 4 4 8 4 k=2  2k ∞ X 59 5 5 {ζ (2k) − 1} = − π; 4 18 8 k=1  2k ∞ X 5 73 {ζ (2k + 1) − 1} = − + 3 log 2; 4 45 k=1  k ∞ X 21 7 293 7 + π− log 2; (−1)k {ζ (k) − 1} = 4 132 8 4 k=2  k ∞ X 7 7 7 21 {ζ (k) − 1} log 2; = + π+ 4 4 8 4 k=2  2k ∞ X 7 131 7 {ζ (2k) − 1} = + π; 4 66 8 k=1  2k ∞ X 7 31 {ζ (2k + 1) − 1} =− + 3 log 2; 4 231 k=1 r √ √ ∞ X 29 π 2√ 1 5 1+ 5 k ζ (k) − 1 (−1) = − 1+ 5 − log 5 − log ; 30 10 5 4 10 2 5k k=2 r √ √ ∞ X ζ (k) − 1 1 π 2√ 1 5 1+ 5 =− − 1+ 5 + log 5 + log ; 20 10 5 4 10 2 5k k=2

287

(239)

(240)

(241)

(242)

(243)

(244)

(245)

(246)

(247)

(248)

(249)

(250)

(251)

(252)

(253)

(254)

288

Zeta and q-Zeta Functions and Associated Series and Integrals ∞ X ζ (2k) − 1 k=1 ∞ X

52k

11 π = − 24 10

r 1+

2√ 5; 5

√ √ 61 5 5 1+ 5 ζ (2k + 1) − 1 = − + log 5 + log ; 24 4 2 2 52k k=1  k ∞ X 2 k (−1) {ζ (k) − 1} 5 k=2 r √ √ 2√ 1 31 π 5 1+ 5 1− 5 − log 5 + = − log ; 35 5 5 2 5 2  k ∞ X 2 {ζ (k) − 1} 5 k=2 r √ √ 4 π 2√ 1 5 1+ 5 =− − 1− 5 + log 5 − log ; 15 5 5 2 5 2 r  2k ∞ X 2√ 2 13 π 1− 5; {ζ (2k) − 1} = − 5 42 5 5 k=1 √ √  2k ∞ X 121 5 5 1+ 5 2 {ζ (2k + 1) − 1} =− + log 5 − log ; 5 84 4 2 2 k=1  k ∞ X 3 (−1)k {ζ (k) − 1} 5 k=2 r √ √ 2√ 3 3 5 1+ 5 31 3π + 1− 5 − log 5 + log ; = 40 10 5 4 10 2  k ∞ X 3 {ζ (k) − 1} 5 k=2 r √ √ 2√ 3 9 3π 3 5 1+ 5 1− 5 + log 5 − =− + log ; 10 10 5 4 10 2 r  2k ∞ X 3 1 3π 2√ {ζ (2k) − 1} =− + 1− 5; 5 16 10 5 k=1 √ √  2k ∞ X 67 5 5 1+ 5 3 = − + log 5 − log ; {ζ (2k + 1) − 1} 5 48 4 2 2 k=1  k ∞ X 4 (−1)k {ζ (k) − 1} 5 k=2 r √ √ 29 2π 2√ 2 5 1+ 5 = + 1+ 5 − log 5 − log ; 45 5 5 5 2

(255)

(256)

(257)

(258)

(259)

(260)

(261)

(262)

(263)

(264)

(265)

Series Involving Zeta Functions

 k ∞ X 4 {ζ (k) − 1} 5 k=2 r √ √ 1+ 5 2√ 2 5 16 2π log ; 1+ 5 + log 5 + =− + 5 5 5 5 2 r   ∞ X 4 2k 23 2π 2√ {ζ (2k) − 1} =− + 1+ 5; 5 18 5 5 k=1 √ √  2k ∞ X 4 173 5 5 1+ 5 {ζ (2k + 1) − 1} =− + log 5 + log ; 5 72 4 2 2 k=1  k ∞ X 6 (−1)k {ζ (k) − 1} 5 k=2 r √ √ 349 3π 2√ 3 3 5 1+ 5 = − 1+ 5 − log 5 − log ; 55 5 5 2 5 2  k ∞ X 6 {ζ (k) − 1} 5 k=2 r √ √ 2√ 3 3 5 1+ 5 6 3π 1+ 5 + log 5 + log ; = − 5 5 5 2 5 2 r   ∞ X 6 2k 83 3π 2√ {ζ (2k) − 1} 1+ 5; = − 5 22 5 5 k=1 √ √  2k ∞ X 283 5 5 1+ 5 6 {ζ (2k + 1) − 1} =− + log 5 + log ; 5 132 4 2 2 k=1  k ∞ X 7 (−1)k {ζ (k) − 1} 5 k=2 r √ √ 2√ 7 221 7π 7 5 1+ 5 1− 5 − log 5 + = − log ; 60 10 5 4 10 2  k ∞ X 7 {ζ (k) − 1} 5 k=2 r √ √ 2√ 7 7 5 1+ 5 7 7π = − 1− 5 + log 5 − log ; 5 10 5 4 10 2 r   ∞ X 7 2k 61 7π 2√ {ζ (2k) − 1} = − 1− 5; 5 24 10 5 k=1 √ √  2k ∞ X 7 137 5 5 1+ 5 {ζ (2k + 1) − 1} =− + log 5 − log ; 5 168 4 2 2 k=1

289

(266)

(267)

(268)

(269)

(270)

(271)

(272)

(273)

(274)

(275)

(276)

290

Zeta and q-Zeta Functions and Associated Series and Integrals

 k ∞ X 8 (−1)k {ζ (k) − 1} 5 k=2 r √ √ 1+ 5 2√ 4 5 523 4π + log ; 1− 5 − 2 log 5 + = 195 5 5 5 2   ∞ X 8 k {ζ (k) − 1} 5 k=2 r √ √ 1+ 5 8 4π 2√ 4 5 log ; = + 1− 5 + 2 log 5 − 5 5 5 5 2 r  2k ∞ X 8 167 4π 2√ {ζ (2k) − 1} = + 1− 5; 5 78 5 5 k=1 √ √  2k ∞ X 211 5 5 1+ 5 8 {ζ (2k + 1) − 1} =− + log 5 − log ; 5 624 4 2 2 k=1  k ∞ X 9 (−1)k {ζ (k) − 1} 5 k=2 r √ √ 2√ 9 9 5 1+ 5 293 9π + 1+ 5 − log 5 − log ; = 140 10 5 4 10 2   ∞ X 9 k {ζ (k) − 1} 5 k=2 r √ √ 2√ 9 9 5 1+ 5 9 9π 1+ 5 + log 5 + log ; = + 5 10 5 4 10 2 r  2k ∞ X 9 2√ 109 9π {ζ (2k) − 1} + 1+ 5; = 5 56 10 5 k=1 √ √  2k ∞ X 9 41 5 5 1+ 5 {ζ (2k + 1) − 1} =− + log 5 + log ; 5 504 4 2 2 k=1 ∞ X

(−1)k

k=2 ∞ X k=2 ∞ X k=1

ζ (k) − 1 41 π √ 1 1 = − 3 − log 3 − log 2; k 42 12 4 3 6

(278)

(279)

(280)

(281)

(282)

(283)

(284)

(285)

ζ (k) − 1 1 π √ 1 1 =− − 3 + log 3 + log 2; k 30 12 4 3 6

(286)

ζ (2k) − 1 33 π √ = − 3; 70 12 62k

(287)

∞ X ζ (2k + 1) − 1 k=1

(277)

62k

=−

106 3 + log 3 + 2 log 2; 35 2

(288)

Series Involving Zeta Functions

 k ∞ X 5 41 5π √ 5 5 (−1)k {ζ (k) − 1} = + 3 − log 3 − log 2; 6 66 12 4 3 k=2   ∞ X 25 5π √ 5 5 5 k =− + 3 + log 3 + log 2; {ζ (k) − 1} 6 6 12 4 3 k=2   ∞ X 5 2k 107 5π √ {ζ (2k) − 1} =− + 3; 6 66 12 k=1  2k ∞ X 158 3 5 {ζ (2k + 1) − 1} =− + log 3 + 2 log 2; 6 55 2 k=1   ∞ X 7 k 575 7π √ 7 7 (−1)k {ζ (k) − 1} = − 3 − log 3 − log 2; 6 78 12 4 3 k=2   ∞ X 7 7 7 k 7 7π √ {ζ (k) − 1} = − 3 + log 3 + log 2; 6 6 12 4 3 k=2  2k ∞ X 111 7π √ 7 = {ζ (2k) − 1} − 3; 6 26 12 k=1  2k ∞ X 242 3 7 {ζ (2k + 1) − 1} =− + log 3 + 2 log 2; 6 91 2 k=1   ∞ X 11 11 11 k 11297 11π √ 3 − log 3 − log 2; (−1)k {ζ (k) − 1} = + 6 5610 12 4 3 k=2  k ∞ X 11 11 11π √ 11 11 {ζ (k) − 1} = + 3 + log 3 + log 2; 6 6 12 4 3 k=2  2k ∞ X 11 3597 11π √ + {ζ (2k) − 1} = 3; 6 1870 12 k=1  2k ∞ X 11 46 3 + log 3 + 2 log 2; {ζ (2k + 1) − 1} =− 6 935 2 k=1 √ ∞  1  X √  71 π √ 2 k ζ (k) − 1 (−1) = − 2 + 1 − log 2 + log 3 − 2 2 ; 72 16 2 16 23k

291

(289)

(290)

(291)

(292)

(293)

(294)

(295)

(296)

(297)

(298)

(299)

(300)

k=2

(301) ∞ X k=2

√  1  √  1 π √ 2 ζ (k) − 1 = − − 2 + 1 + log 2 − log 3 − 2 2 ; 56 16 2 16 23k

∞ X ζ (2k) − 1 k=1

26k

=

 61 π √ − 2+1 ; 126 16

(302)

(303)

292

Zeta and q-Zeta Functions and Associated Series and Integrals

√ √ 253 2 =− + 4 log 2 − log(3 − 2 2); 6k 63 2 2 k=1   ∞ X 3 k (−1)k {ζ (k) − 1} 8 k=2 √  3  √  3 2 79 3π √ − 2 − 1 − log 2 − log 3 − 2 2 ; = 88 16 2 16  k ∞ X 3 {ζ (k) − 1} 8 k=2  3  √  9 3π √ 3√ =− − 2 − 1 + log 2 + 2 log 3 − 2 2 ; 40 16 2 16  2k ∞   X √ 3 37 3π {ζ (2k) − 1} = − 2−1 ; 8 110 16 k=1 √  2k ∞  X √  247 2 3 {ζ (2k + 1) − 1} =− + 4 log 2 + log 3 − 2 2 ; 8 165 2 k=1   ∞ X 5 k (−1)k {ζ (k) − 1} 8 k=2 √  √  5 79 5π √ 5 2 = + ( 2 − 1) − log 2 − log 3 − 2 2 ; 104 16 2 16  k ∞ X 5 {ζ (k) − 1} 8 k=2 √  √  5 25 5π √ 5 2 =− + ( 2 − 1) + log 2 + log 3 − 2 2 ; 24 16 2 16  2k ∞  X 5 55 5π √ {ζ (2k) − 1} =− + 2−1 ; 8 78 16 k=1 √   ∞  X √  5 2k 281 2 {ζ (2k + 1) − 1} =− + 4 log 2 + log 3 − 2 2 ; 8 195 2 k=1  k ∞ X 7 (−1)k {ζ (k) − 1} 8 k=2 √  7  √  7π √ 7 2 71 + 2 + 1 − log 2 + log 3 − 2 2 ; = 120 16 2 16  k ∞ X 7 {ζ (k) − 1} 8 k=2 √  7  √  49 7π √ 7 2 =− + 2 + 1 + log 2 − log 3 − 2 2 ; 8 16 2 16 ∞ X ζ (2k + 1) − 1

(304)

(305)

(306)

(307)

(308)

(309)

(310)

(311)

(312)

(313)

(314)

Series Involving Zeta Functions

 2k ∞  X 7 83 7π √ {ζ (2k) − 1} =− + 2+1 ; 8 30 16 k=1 √  2k ∞  X √  403 2 7 =− + 4 log 2 − log 3 − 2 2 ; {ζ (2k + 1) − 1} 8 105 2

293

(315)

(316)

k=1

 k ∞ X 9 k (−1) {ζ (k) − 1} 8 k=2

√  9  √  1279 9π √ 9 2 = 2 + 1 − log 2 + − log 3 − 2 2 ; 136 16 2 16  k ∞ X 9 {ζ (k) − 1} 8 k=2 √   9 √  9 2 9 9π √ log 3 − 2 2 ; = − 2 + 1 + log 2 − 8 16 2 16  2k ∞   X 179 9π √ 9 2+1 ; {ζ (2k) − 1} = − 8 34 16 k=1 √  2k ∞  X √  9 563 2 {ζ (2k + 1) − 1} =− + 4 log 2 − log 3 − 2 2 ; 8 153 2

(317)

(318)

(319)

(320)

k=1

 k ∞ X 11 (−1)k {ζ (k) − 1} 8 k=2

√  11  √  11 2 1765 11π √ − 2 − 1 − log 2 − log 3 − 2 2 ; = 456 16 2 16   ∞ k X 11 {ζ (k) − 1} 8 k=2 √   11 √  11 2 11 11π √ − log 3 − 2 2 ; = 2 − 1 + log 2 + 8 16 2 16   ∞ 2k   X 11 299 11π √ {ζ (2k) − 1} − = 2−1 ; 8 114 16 k=1 √  2k ∞  X √  569 2 11 =− + 4 log 2 + log 3 − 2 2 ; {ζ (2k + 1) − 1} 8 627 2

(321)

(322)

(323)

(324)

k=1

 k ∞ X 13 k (−1) {ζ (k) − 1} 8 k=2

√  13  √  2179 13π √ 13 2 = + 2−1 − log 2 − log 3 − 2 2 ; 840 16 2 16

(325)

294

Zeta and q-Zeta Functions and Associated Series and Integrals

 k ∞ X 13 {ζ (k) − 1} 8 k=2

√  13  √  13 13π √ 13 2 2−1 + = + log 2 + log 3 − 2 2 ; 8 16 2 16  2k ∞   X √ 13 443 13π + {ζ (2k) − 1} = 2−1 ; 8 210 16 k=1 √  2k ∞  X √  407 13 2 =− + 4 log 2 + log 3 − 2 2 ; {ζ (2k + 1) − 1} 8 1365 2 k=1  k ∞ X 15 (−1)k {ζ (k) − 1} 8 k=2 √  15  √  2473 15π √ 15 2 = + 2+1 − log 2 + log 3 − 2 2 ; 1288 16 2 16  k ∞ X 15 {ζ (k) − 1} 8 k=2 √   15 √  15 2 15 15π √ + log 2 − log 3 − 2 2 ; = 2+1 + 8 16 2 16  2k ∞   X √ 611 15π 15 {ζ (2k) − 1} = + 2+1 ; 8 322 16 k=1 √   ∞  X √  29 2 15 2k =− + 4 log 2 − log 3 − 2 2 ; {ζ (2k + 1) − 1} 8 2415 2

(326)

(327)

(328)

(329)

(330)

(331)

(332)

k=1 ∞ X

ζ (k) − 1 10k k=2 √ q  √ √  109 π 1 1 5 = − 5 + 2 5 − log 2 − log 5 − log 2 + 5 ; 110 20 5 8 20 ∞ X ζ (k) − 1 10k k=2 √ q  √ √  1 π 1 1 5 =− − 5 + 2 5 + log 2 + log 5 + log 2 + 5 ; 90 20 5 8 20 q ∞ X √ ζ (2k) − 1 π 97 = − 5 + 2 5; 2k 198 20 10 k=1 √ ∞  X ζ (2k + 1) − 1 √  496 5 5 = − + 2 log 2 + log 5 + log 2 + 5 ; 99 4 2 102k (−1)k

k=1

(333)

(334)

(335)

(336)

Series Involving Zeta Functions

 k ∞ X 3 (−1)k {ζ (k) − 1} 10 k=2 √ q  √ √  3 3 3 5 121 3π − 5 − 2 5 − log 2 − log 5 + log 2 + 5 ; = 130 20 5 8 20  k ∞ X 3 {ζ (k) − 1} 10 k=2 √ q  √ √  3π 3 3 3 5 9 5 − 2 5 + log 2 + log 5 − log 2 + 5 ; =− − 70 20 5 8 20 q  2k ∞ X √ 73 3π 3 = − {ζ (2k) − 1} 5 − 2 5; 10 182 20

295

(337)

(338)

(339)

k=1

 2k ∞ X 3 {ζ (2k + 1) − 1} 10 k=1

√  √  482 5 5 =− + 2 log 2 + log 5 − log 2 + 5 ; 273 4 2 q  k ∞ X √ 121 7π 7 (−1)k {ζ (k) − 1} = + 5−2 5 10 170 20 k=2 √  √  7 7 7 5 − log 2 − log 5 + log 2 + 5 ; 5 8 20  k ∞ X 7 {ζ (k) − 1} 10 k=2 √ q  √ √  7 7 5 7 49 7π 5 − 2 5 + log 2 + log 5 − =− + log 2 + 5 ; 30 20 5 8 20 q  2k ∞ X √ 7 47 7π {ζ (2k) − 1} =− + 5 − 2 5; 10 102 20

(340)

(341)

(342)

(343)

k=1

 2k ∞ X 7 {ζ (2k + 1) − 1} 10 k=1

√  √  598 5 5 =− + 2 log 2 + log 5 − log 2 + 5 ; 357 4 2 q  k ∞ X √ 9 91 9π (−1)k {ζ (k) − 1} = + 5+2 5 10 190 20 k=2 √  √  9 9 9 5 − log 2 − log 5 − log 2 + 5 ; 5 8 20

(344)

(345)

296

Zeta and q-Zeta Functions and Associated Series and Integrals

 k ∞ X 9 {ζ (k) − 1} 10 k=2 √ q  √ √  9 9 9 5 81 9π 5 + 2 5 + log 2 + log 5 + log 2 + 5 ; =− + 10 20 5 8 20 q  2k ∞ X √ 9 143 9π {ζ (2k) − 1} 5 + 2 5; =− + 10 38 20 k=1  2k ∞ X 9 {ζ (2k + 1) − 1} 10 k=1 √  √  5 5 824 =− + 2 log 2 + log 5 + log 2 + 5 ; 171 4 2 q  k ∞ X √ 2399 11π 11 k = − 5+2 5 (−1) {ζ (k) − 1} 10 210 20 k=2 √  √  11 11 11 5 − log 2 − log 5 − log 2 + 5 ; 5 8 20 q  k ∞ X √ 11 11π 11 = − 5+2 5 {ζ (k) − 1} 10 10 20 k=2 √  √  11 11 11 5 + log 2 + log 5 + log 2 + 5 ; 5 8 20 q  2k ∞ X √ 11 263 11π {ζ (2k) − 1} = − 5 + 2 5; 10 42 20 k=1   ∞ X 11 2k {ζ (2k + 1) − 1} 10 k=1 √  √  5 1084 5 =− + 2 log 2 + log 5 + log 2 + 5 ; 231 4 2 q  k ∞ X √ 13 3173 13π − (−1)k {ζ (k) − 1} = 5−2 5 10 690 20 k=2 √  √  13 13 13 5 − log 2 − log 5 + log 2 + 5 ; 5 8 20 q  k ∞ X √ 13 13 13π {ζ (k) − 1} = − 5−2 5 10 10 20 k=2 √  √  13 13 13 5 + log 2 + log 5 − log 2 + 5 ; 5 8 20 q  2k ∞ X √ 13 407 13π {ζ (2k) − 1} = − 5 − 2 5; 10 138 20 k=1

(346)

(347)

(348)

(349)

(350)

(351)

(352)

(353)

(354)

(355)

Series Involving Zeta Functions

297

 2k ∞ X 13 {ζ (2k + 1) − 1} 10 k=1

√  √  5 5 1138 + 2 log 2 + log 5 − log 2 + 5 ; =− 897 4 2 q   ∞ k X √ 17 4457 17π (−1)k {ζ (k) − 1} = + 5−2 5 10 1890 20 k=2 √  √  17 17 17 5 − log 2 − log 5 + log 2 + 5 ; 5 8 20 q   ∞ X √ 17 k 17 17π {ζ (k) − 1} = + 5−2 5 10 10 20 k=2 √  √  17 17 17 5 + log 2 + log 5 − log 2 + 5 ; 5 8 20 q   ∞ 2k X √ 767 17π 17 = + {ζ (2k) − 1} 5 − 2 5; 10 378 20

(356)

(357)

(358)

(359)

k=1

 2k ∞ X 17 {ζ (2k + 1) − 1} 10 k=1

√  √  622 5 5 =− + 2 log 2 + log 5 − log 2 + 5 ; 3213 4 2 q   ∞ X √ 19 k 4871 19π (−1)k {ζ (k) − 1} 5+2 5 = + 10 261 20 k=2 √  √  19 19 19 5 − log 2 − log 5 − log 2 + 5 ; 5 8 20 q   ∞ k X √ 19 19π 19 + 5+2 5 {ζ (k) − 1} = 10 20 20 k=2 √  √  19 19 19 5 + log 2 + log 5 + log 2 + 5 ; 5 8 20 q   ∞ X √ 19 2k 983 19π {ζ (2k) − 1} = + 5 + 2 5; 10 522 20

(360)

(361)

(362)

(363)

k=1

 2k ∞ X 19 {ζ (2k + 1) − 1} 10 k=1

√  √  44 5 5 = + 2 log 2 + log 5 + log 2 + 5 ; 4959 4 2

(364)

298

Zeta and q-Zeta Functions and Associated Series and Integrals ∞ X ζ (k) − 1 (−1)k 12k k=2

√  √ √  1 3 155 π 1 = − (2 + 3) + log 7 − 4 3 − log 3 − log 2; (365) 156 24 24 8 4 ∞ X ζ (k) − 1 12k k=2 (366) √  √  √  1 1 π  3 1 =− − 2+ 3 − log 7 − 4 3 + log 3 + log 2; 132 r24 24 8 4 ∞ X √  π  ζ (2k) − 1 141 = − 2+ 3 ; (367) 286 24 122k k=1 √ ∞  X √  3 ζ (2k + 1) − 1 859 3 − log 7 − 4 (368) = − 3 + log 3 + 3 log 2; 143 2 2 122k k=1  k ∞ X 5 k (−1) {ζ (k) − 1} 12 k=2 √  √  5 3 √  5 179 5π  5 = − 2− 3 − log 7 − 4 3 − log 3 − log 2; (369) 204 24 24 8 4  k ∞ X 5 {ζ (k) − 1} 12 k=2 √  √  5 3 √  5 25 5π  5 =− − 2− 3 + log 7 − 4 3 + log 3 + log 2; (370) 84 24 24 8 4   ∞ X √  69 5π  5 2k = − 2− 3 ; (371) {ζ (2k) − 1} 12 238 24 k=1 √  2k ∞  X √  3 5 839 3 + log 7 − 4 3 + log 3 + 3 log 2; {ζ (2k + 1) − 1} =− 12 595 2 2 k=1

(372) ∞ X

(−1)k {ζ (k) − 1}

k=2



7 12

k

√  √  7 3 √  7 179 7π  7 = + 2− 3 − log 7 − 4 3 − log 3 − log 2; 228 24 24 8 4  k ∞ X 7 {ζ (k) − 1} 12 k=2 √  √  7 3 √  7 49 7π  7 =− + 2− 3 + log 7 − 4 3 + log 3 + log 2; 60 24 24 8 4

(373)

(374)

Series Involving Zeta Functions

299

 2k ∞ X √  7 3 7π  {ζ (2k) − 1} =− + 2− 3 ; (375) 12 190 24 k=1 √  2k ∞  X √  3 913 3 7 =− + log 7 − 4 3 + log 3 + 3 log 2; {ζ (2k + 1) − 1} 12 665 2 2 k=1

(376) ∞ X

(−1)k {ζ (k) − 1}

k=2



11 12

k

(377) √  √  11 3 √  11 155 11π  11 = + 2+ 3 + log 7 − 4 3 − log 3 − log 2; 276 24 24 8 4   ∞ k X 11 {ζ (k) − 1} 12 k=2 (378) √  √  11 3 √  11 121 11π  11 =− + 2+ 3 − log 7 − 4 3 + log 3 + log 2; 12 24 24 8 4  2k ∞   X √ 11 219 11π {ζ (2k) − 1} (379) =− + 2+ 3 ; 12 46 24 k=1

 2k ∞ X 11 {ζ (2k + 1) − 1} 12 k=1 (380) √  √  3 1469 3 =− − log 7 − 4 3 + log 3 + 3 log 2; 253 2 2  k ∞ X 13 (−1)k {ζ (k) − 1} 12 k=2 (381) √  √ √  13 4031 13π 13 3 13 = − (2 + 3) + log 7 − 4 3 − log 3 − log 2; 300 24 24 8 4  k ∞ X 13 {ζ (k) − 1} 12 k=2 (382) √  √  13 3 √  13 13 13π  13 2+ 3 − = − log 7 − 4 3 + log 3 + log 2; 12 24 24 8 4  2k ∞ X √  13 363 13π  (383) {ζ (2k) − 1} = − 2+ 3 ; 12 50 24 k=1 √  2k ∞  X √  3 13 1853 3 {ζ (2k + 1) − 1} =− − log 7 − 4 3 + log 3 + 3 log 2; 12 325 2 2 k=1

(384)

300

Zeta and q-Zeta Functions and Associated Series and Integrals

 k ∞ X 17 (−1)k {ζ (k) − 1} 12 k=2

(385) √  √  17 3 √  17 17 6211 17π  − 2− 3 − log 7 − 4 3 − log 3 − log 2; = 1740 24 24 8 4   ∞ X 17 k {ζ (k) − 1} 12 k=2 (386) √  √ √  17 17 17 3 17 17π − (2 − 3) + log 7 − 4 3 + log 3 + log 2; = 12 24 24 8 4  2k ∞ X √  723 17π  17 = − 2− 3 ; {ζ (2k) − 1} (387) 12 290 24 k=1 √  2k ∞  X √  3 1873 17 3 =− + log 7 − 4 3 + log 3 + 3 log 2; {ζ (2k + 1) − 1} 12 2465 2 2 k=1

(388) ∞ X

(−1)k {ζ (k) − 1}

k=2



19 12

k

(389) √  √ √  19 19 7145 19π 19 3 + (2 − 3) − log 7 − 4 3 − log 3 − log 2; = 2604 24 24 8 4  k ∞ X 19 {ζ (k) − 1} 12 k=2 (390) √  √  19 3 √  19 19 19π  19 = + 2− 3 + log 7 − 4 3 + log 3 + log 2; 12 24 24 8 4  2k ∞ X √  939 19π  19 = + 2− 3 ; (391) {ζ (2k) − 1} 12 434 24 k=1 √  2k ∞  X √  3 19 1511 3 + log 7 − 4 3 + log 3 + 3 log 2; {ζ (2k + 1) − 1} =− 12 4123 2 2 k=1

(392)  k ∞ X 23 (−1)k {ζ (k) − 1} 12 k=2

(393) √  √ √  23 8461 23π 23 3 23 = + (2 + 3) + log 7 − 4 3 − log 3 − log 2; 4620 24 24 8 4  k ∞ X 23 {ζ (k) − 1} 12 k=2 (394) √  √ √  23 23 23π 23 3 23 = + (2 + 3) − log 7 − 4 3 + log 3 + log 2; 12 24 24 8 4

Series Involving Zeta Functions

301

 2k ∞ X √  23 1443 23π  {ζ (2k) − 1} = + 2+ 3 ; (395) 12 770 24 k=1 √  2k ∞  X √  3 197 3 23 = − log 7 − 4 3 + log 3 + 3 log 2. {ζ (2k + 1) − 1} 12 8855 2 2 k=1

(396) Setting a = 21 in (1) through (8) and using 1.1(14), 2.3(2) and appropriate special values of the ψ-function listed in Section 1.3, we obtain   ∞ X 1 tk 1 − t − (γ + 2 log 2)t − log π (2k − 1)ζ (k) = log 0 k 2 2



|t| <

k=2

 1 ; 2

(397)     ∞ X tk 1 1 1 (−1)k (2k − 1)ζ (k) = log 0 + t + (γ + 2 log 2)t − log π |t| < ; k 2 2 2 k=2

    ∞   X t2k 1 1 22k − 1 ζ (2k) = log 0 + t + log 0 − t − log π k 2 2 k=1

(398)   1 |t| < ; 2 (399)

∞   X t2k+1 22k+1 − 1 ζ (2k + 1) 2k + 1 k=1      1 1 1 = log 0 − t − log 0 + t − (γ + 2 log 2)t 2 2 2    ∞ X 1 k k−1 (2 − 1)ζ (k)t = −ψ − t − γ − 2 log 2 |t| < 2 k=2

  ∞ X 1 k k k−1 (−1) (2 − 1)ζ (k)t =ψ + t + γ + 2 log 2 2 k=2

  1 |t| < ; 2

     ∞   X 1 1 1 +t −ψ −t 22k − 1 ζ (2k)t2k−1 = ψ 2 2 2 k=1

  1 |t| < ; 2  1 ; 2

  1 |t| < ; 2

(400)

(401)

(402)

(403)

∞   X 22k+1 − 1 ζ (2k + 1)t2k k=1

     1 1 1 =− ψ +t +ψ − t − γ − 2 log 2 2 2 2

  1 |t| < . 2

(404)

302

Zeta and q-Zeta Functions and Associated Series and Integrals

For z = t + 12 , 1.1(12) yields    1 1 π 0 +t 0 −t = , 2 2 cos(πt) 

(405)

which can be used to rewrite the series identity (399) in its equivalent form: ∞   X t2k = − log cos(π t) 22k − 1 ζ (2k) k k=1

  1 |t| < . 2

(406)

Furthermore, in view of 1.3(11), the series identity (403) can be rewritten in its equivalent form: ∞   X π 22k − 1 ζ (2k)t2k−1 = tan(πt) 2 k=1

  1 |t| < . 2

(407)

For suitable special values of the argument t, we can easily deduce the following series identities from (397) through (404), (406) and (407):   ∞ X 1 ζ (k) 1 1 k (−1) 1 − k = γ − log π + log 2; k 2 2 2 k=2 ∞ 2k X 2

−1 ζ (2k) = log 2; k · 32k

(409)

−1 1 ζ (2k) = log 2; 2k 2 k·4

(410)

−1 1 ζ (2k) = log 2 − log 3; 2 k · 62k

(411)

k=1 ∞ 2k X 2 k=1 ∞ 2k X 2 k=1 ∞ 2k X 2

 √  −1 1 ζ (2k) = log 4 − 2 2 ; 2k 2 k·8 k=1  2k ∞ 2k  X √  3 2 −1 1 ζ (2k) = log 4 + 2 2 ; k 8 2 k=1 ∞ 2k X 2

√ √  −1 ζ (2k) = log 6 − 2 ; k · 122k k=1  2k ∞ 2k √ X √  2 −1 5 ζ (2k) = log 6+ 2 ; k 12 k=1   ∞ X 1 (−1)k 1 − k ζ (k) = log 2; 2 k=2

(408)

(412)

(413)

(414)

(415)

(416)

Series Involving Zeta Functions

303

∞ X 2k − 1 π√ 1 (−1)k k ζ (k) = 3 − log 3; 6 2 3

(417)

π√ 1 −1 ζ (k) = 3 + log 3; k 6 2 3

(418)

k=2 ∞ k X 2 k=2

∞ 2k X 2 −1

32k

k=1

ζ (2k) =

∞ 2k+1 X 2 −1

32k+1

k=1

π√ 3; 6

ζ (2k + 1) =

1 log 3; 2

∞ X 2k − 1 π 1 (−1)k k ζ (k) = − log 2; 8 4 4

(419)

(420)

(421)

k=2

∞ k X 2 −1

4k

k=2

ζ (k) =

∞ 2k X 2 −1

42k

k=1

ζ (2k) =

∞ 2k+1 X 2 −1 k=1 ∞ X

42k

π 1 + log 2; 8 4 π ; 8

ζ (2k + 1) = log 2;

2k − 1 ζ (k) 5k k=2 √ q  √ √  π 1 5 = 5 − 2 5 − log 5 + log 2 + 5 ; 10 4 10 ∞ k X 2 −1 ζ (k) 5k k=2 √ q  √ √  π 1 5 = 5 − 2 5 + log 5 − log 2 + 5 ; 10 4 10 q ∞ 2k X √ 2 −1 π ζ (2k) = 5 − 2 5; 2k 10 5 k=1 √ ∞  X 22k+1 − 1 √  5 5 ζ (2k + 1) = log 5 − log 2 + 5 ; 4 2 52k k=1  k ∞   X 2 (−1)k 2k − 1 ζ (k) 5 k=2 √ q  √ √  π 1 5 = 5 + 2 5 − log 5 − log 2 + 5 ; 5 2 5

(422)

(423)

(424)

(−1)k

(425)

(426)

(427)

(428)

(429)

304

Zeta and q-Zeta Functions and Associated Series and Integrals

√ q  k ∞    X √ √  2 π 1 5 k 2 − 1 ζ (k) = 5 + 2 5 + log 5 + log 2 + 5 ; 5 5 2 5

(430)

q  2k ∞   X √ π 2 2k 5 + 2 5; = 2 − 1 ζ (2k) 5 5

(431)

√  2k ∞    X √  2 5 5 2k+1 2 − 1 ζ (2k + 1) = log 5 + log 2 + 5 ; 5 4 2

(432)

k=2

k=1

k=1

∞ X π √ 1 1 2k − 1 3 − log 3 + log 2; (−1)k k ζ (k) = 36 4 3 6

(433)

k=2

∞ k X 2 −1 k=2

6k

ζ (k) =

∞ 2k X 2 −1 k=1

k · 52k

∞ 2k X 2 −1 k=1

k

1 π √ 1 3 + log 3 − log 2; 36 4 3

ζ (2k) = − log

√ 5+1 ; 4

√  2k 5−1 2 ζ (2k) = − log ; 5 4

√ 2 2 ζ (2k) = log p √ ; k · 102k 5+ 5 k=1 √  2k ∞ 2k X 2 −1 2 2 3 = log p ζ (2k) √ ; k 10 5 − 5 k=1

∞ 2k X 2 −1

∞ 2k X 2 −1 k=1

62k

ζ (2k) =

∞ 2k+1 X 2 −1 k=1

62k+1

π √ 3; 36

ζ (2k + 1) =

1 1 log 3 − log 2; 4 3

∞  √2  k X √  1 π √ k2 −1 (−1) ζ (k) = 2 − 1 − log 3 − 2 2 − log 2; 16 16 4 8k

(434)

(435)

(436)

(437)

(438)

(439)

(440)

(441)

k=2

∞ k X 2 −1 k=2

8k

 √2  √  1 π √ ζ (k) = 2−1 + log 3 − 2 2 + log 2; 16 16 4

∞ 2k X 2 −1 k=1

82k

ζ (2k) =

 π √ 2−1 ; 16

(442)

(443)

Series Involving Zeta Functions ∞ 2k+1 X 2 −1 k=1

82k+1

√  √  1 2 ζ (2k + 1) = log 2 + log 3 − 2 2 ; 4 16

305

(444)

√  k ∞  3    X √  3 3π √ 3 2 k k 2 + 1 − log 2 + = log 3 − 2 2 ; (−1) 2 − 1 ζ (k) 8 16 4 16 k=2

(445) √  k ∞    3  X √  3 3π √ 3 2 k 2 − 1 ζ (k) = 2 + 1 + log 2 − log 3 − 2 2 ; 8 16 4 16 k=2

(446)  2k ∞    X 3 3π √ 22k − 1 ζ (2k) = 2+1 ; 8 16

(447)

√  2k ∞    X √  3 2 22k+1 − 1 ζ (2k + 1) = 2 log 2 − log 3 − 2 2 ; 8 2

(448)

k=1

k=1

∞ X 2k − 1 (−1)k ζ (k) 10k k=2 r √ √ π 2√ 1 1 5 1+ 5 = 1− 5 + log 2 − log 5 + log ; 20 5 5 8 20 2 ∞ k X 2 −1 ζ (k) 10k k=2 r √ √ π 2√ 1 1 1+ 5 5 = 1− 5 − log 2 + log 5 − log ; 20 5 5 8 20 2 r ∞ 2k X 2 −1 2√ π ζ (2k) = 1 − 5; 20 5 102k k=1 √ √ ∞ 2k+1 X 2 −1 5 5 1+ 5 ζ (2k + 1) = −2 log 2 + log 5 − log ; 4 2 2 102k k=1  k ∞   X 3 k k (−1) 2 − 1 ζ (k) 10 k=2 r √ √ 2√ 3 3 3 5 1+ 5 3π 1+ 5 + log 2 − log 5 − log ; = 20 5 5 8 20 2  k ∞   X 3 2k − 1 ζ (k) 10 k=2 r √ √ 3π 3 3 5 1+ 5 2√ 3 = log ; 1+ 5 − log 2 + log 5 + 20 5 5 8 20 2

(449)

(450)

(451)

(452)

(453)

(454)

306

Zeta and q-Zeta Functions and Associated Series and Integrals

r  2k ∞   X 3 3π 2√ 2k 2 − 1 ζ (2k) = 1+ 5; (455) 10 20 5 k=1 √ √  2k ∞   X 5 1+ 5 3 5 2k+1 = −2 log 2 + log 5 + log ; 2 − 1 ζ (2k + 1) 10 4 2 2 k=1

(456) √ √ √ 2k − 1 π 1 1 3 (−1)k ζ (k) = (2 − 3) − log(7 − 4 3) − log 3 − log 2; k 24 24 8 12 12

∞ X k=2

√  √ √  1 π 1 −1 3 ζ (k) = (2 − log 7 − 4 log 2; 3) + 3 + log 3 + k 24 24 8 12 12

(457)

∞ k X 2 k=2

(458) ∞ 2k X 2 −1

√  π  ζ (2k) = 2 − 3 ; 24 122k k=1 √ ∞ 2k+1  X √  1 3 1 2 −1 ζ (2k + 1) = log 7 − 4 3 + log 3 + log 2; 2k+1 24 8 12 12 k=1 √  k ∞    X √  5 3 √  5 5π  k k (−1) 2 − 1 ζ (k) = 2+ 3 + log 7 − 4 3 12 24 24 k=2

5 5 log 3 − log 2; 8 12 √ ∞    X √  5 3 √  5π  2+ 3 − log 7 − 4 3 2k − 1 ζ (k) (512)k = 24 24

(459) (460)

(461)

−

k=2

(462) 5 5 log 3 + log 2; 8 12  2k ∞    X √  5 5π 22k − 1 ζ (2k) 2+ 3 ; (463) = 12 24 k=1 √  2k ∞    X √  3 3 5 2k+1 2 − 1 ζ (2k + 1) =− log 7 − 4 3 + log 3 + log 2. 12 2 2 +

k=1

(464)

Evaluation by Using the Double and Triple Gamma Functions We first derive some useful integral expressions for log 0(z + a), log G(z + a) and log 03 (z + a). Setting z = t + a − 1 in 1.1(2) and 1.4(3) and taking the logarithmic derivatives of the resulting equations, we obtain ∞  X k=1

1 1 − t+a−1+k k

 =−

0 0 (t + a) −γ 0(t + a)

(465)

Series Involving Zeta Functions

307

and ∞  X k=1

t+a−1 k −1+ t+a−1+k k

 =

G0 (t + a) 1 − log(2π) G(t + a) 2

(466)

1 + + (1 + γ )(t + a − 1), 2 respectively. Now, we set z = t + a − 1 in 1.4(90) and take the logarithmic derivative of the resulting equation. We, thus, find that 030 (t + a) = 03 (t + a)



 3 1 − log(2π) − log A 8 4     1 1 1 π2 3 + γ + log(2π) + (t + a − 1) − γ+ + (t + a − 1)2 2 2 2 6 2 "∞   k t+a−1 t+a−1 X −1+ + 2 t+a−1+k rk k=1 #   ∞ X 1 1 π2 + − + (t + a − 1) . (467) t+a−1+k k 6 k=1

A combination of (465) and (466) yields 0 0 (t + a) G0 (t + a) 1 1 = + log(2π) − a − t + (t + a − 1) . G(t + a) 2 2 0(t + a)

(468)

Equation (467), by virtue of (465) and (466), becomes 030 (t + a) = 03 (t + a)



 3 1 − log(2π) − log A 8 4   1 1 (t + a − 1)2 + + log(2π) (t + a − 1) − 2 4 4 1 G0 (t + a) 1 0 0 (t + a) + (t + a − 1) − (t + a − 1) . 2 G(t + a) 2 0(t + a)

(469)

Integrating both sides of (468) from t = 0 to t = z, we have Zz

log 0(t + a)dt

0

 1 1 z2 = + log(2π) − a z − + (z + a − 1) log 0(z + a) 2 2 2 − log G(z + a) + (1 − a) log 0(a) + log G(a), 

(470)

308

Zeta and q-Zeta Functions and Associated Series and Integrals

which is due to Barnes [94, p. 283] and, in the special case when a = 1, reduces at once to Alexeiewsky’s theorem 1.4(42). Integrating both sides of (469) with respect to t from t = 0 to t = z with the aid of Barnes’s integral formula (470), we obtain Zz

 log G(t + a)dt =

 a2 1 1 z (a − 1) log(2π) − 2 log A − +a− 2 2 4

0

1 1 [log(2π) + 2 − 2a] z2 − z3 4 6 + (z + a − 2) log G(z + a) − 2 log 03 (z + a) + (2 − a) log G(a) + 2 log 03 (a). +

(471)

In addition to 1.4(73) and 1.4(77), we can also evaluate each of the following integrals by direct use of the triple Gamma function 03 in 1.4(90): Z1

log 03 (t + 1)dt = −

1 3 log(2π) + log B 24 2

log 03 (t + 1)dt = −

1 29 1 − log 2 − log π 256 1920 48

(472)

0

and 1

Z2 0

+

(473)

1 3 15 log A + log B + log C. 16 4 16

For example, to evaluate the integral in (473), we take the logarithms on both sides of the equation 1.4(90) and integrate the resulting equation from t = 0 to t = 12 . We then obtain 1

Z2

log 03 (t + 1)dt =

γ 1 1 37 + − log(2π) − log A 768 128 48 8

0

+

1 lim Sn , 16 n→∞

where Sn : =

n  X

n o − (2k + 1)3 log(2k + 1) + (2k)3 log(2k)

k=1

+ {(2k + 1) log(2k + 1) + (2k) log(2k)}  5 1 +(16k + 12k + 2k) log(2k) + (4k + 5k) + − , 6 8k 3

2

2

(474)

Series Involving Zeta Functions

309

which immediately yields

Sn = −

2n+1 X

k3 log k +

k=1 n X

+2

2n+1 X

k log k + 16

k=1

k=1

+ (5 + 2 log 2)

k=1

k−

k3 log k + 12

k=1

k log k + 16 log 2 n X

n X

1 8

n X

k3 + (4 + 12 log 2)

k=1 n X k=1

n X

k2 log k

k=1 n X 2

k

k=1

1 5 + n. k 6

Using 1.1(3), 1.3(2), 1.3(69) and 1.3(70), we have

Sn =

11 1 log 2 − γ + 3 log A + 12 log B + 15 log C 120 8     1 11 log 1 + − 4n4 + 12n3 + 11n2 + 3n − 120 2n   11 25 13 1 + 2n3 + n2 + n − +O (n → ∞), 2 6 48 n

which, by means of the asymptotic formula 1.3(76), yields 5 11 γ lim Sn = − + log 2 − + 3 log A + 12 log B + 15 log C. 6 120 8

n→∞

(475)

Finally, by substituting (475) into (474), we obtain the desired formula (473). We shall show that integrals of the forms: Zz

tk ψ(t + a)dt (k ∈ N)

0

can be expressed in terms of multiple Gamma functions. First of all, integrating by parts with the aid of (470), we obtain Zz

1 z2 tψ(t + a)dt = [ 2a − 1 − log(2π) ]z + + (1 − a) log 0(z + a) 2 2

0

+ log G(z + a) + (a − 1) log 0(a) − log G(a).

(476)

310

Zeta and q-Zeta Functions and Associated Series and Integrals

Conversely, integrating by parts with the aid of (470) and (471), we obtain Zz 2

t log 0(t + a)dt =

0



   1 2 1 a 1 2 1 1 − a + a − 2 log A z + log(2π) − + z 4 2 2 2 2 4 (477)

z3 + (z2 − (a − 1)2 ) log 0(z + a) + (2a − 3) log G(z + a) 2 − 2 log 03 (z + a) + (a − 1)2 log 0(a) + (3 − 2a) log G(a) + 2 log 03 (a). −

Next, integrating by parts with the aid of (477), we obtain Zz

t2 ψ(t + a)dt =

a−1 z2 z3 {1 − 2a + log(2π)} z + {1 − log(2π)} + 2 4 3

0

+ (a − 1)2 log 0(a + z) + (z − a + 1) log G(a + z) Zz 2 − (a − 1) log 0(a) + (a − 1) log G(a) − log G(t + a) dt 0

(478)   1 2 1 1 = − + a − a + 2 log A z 4 2 2   1 a 1 2 z3 + − log(2π) + − z + + (a − 1)2 log 0(z + a) 2 2 4 2 + (3 − 2a) log G(z + a) + 2 log 03 (z + a) − (a − 1)2 log 0(a) + (2a − 3) log G(a) − 2 log 03 (a). Furthermore, integrating by parts with the aid of (471), we obtain Zz 2

  1 1 a2 t log G(t + a)dt = (2 − a) − + (a − 1) log(2π) − 2 log A − +a z 4 2 2

0

  1 7 1 a2 + log(2π) − 2 log A + − 2a z2 2 4 2 2 1 1 + (log(2π) − a)z3 − z4 + (z2 − a2 + 4a − 4) log G(z + a) 6 8 + 2(a − 2 − z) log 03 (z + a) + (a − 2)2 log G(a) Zz + 2(2 − a) log 03 (a) + 2 log 03 (t + a)dt. +

0

(479)

Series Involving Zeta Functions

311

Finally, integrating by parts with the aid of (477) and (479), we obtain Zz 3

t2 log 0(t + a)dt

0

  1 1 3 = −a2 + a − + 2(2a − 3) log A − (a2 − 3a + 2) log(2π) z 2 4 2   9 7 3 1 − a + a2 + (3 − 2a) log(2π) − log A z2 + 8 4 4 4 n o 1 3 + [log(2π) − 1]z3 − z4 + z3 + (a − 1)3 log 0(z + a) 3 8

(480)

− (3a2 − 9a + 7) log G(z + a) + 2(2a − 3 − z) log 03 (z + a) + (1 − a)3 log 0(a) + (3a − 9a + 7) log G(a) + 2(3 − 2a) log 03 (a) + 2 2

Zz

log 03 (t + a)dt

0

and Zz

t3 ψ(t + a)dt

0

  1 1 2 3 2 = a − a + + 2(3 − 2a) log A + (a − 3a + 2) log(2π) z 2 4 2   3 2 7 9 1 + − a + a − + (2a − 3) log(2π) + log A z2 4 4 8 4 3 1 + [1 − log(2π)]z3 + z4 + (1 − a)3 log 0(z + a) 3 8

(481)

+ (3a2 − 9a + 7) log G(z + a) + 2(z − 2a + 3) log 03 (z + a) + (a − 1)3 log 0(a) − (3a − 9a + 7) log G(a) + 2(2a − 3) log 03 (a) − 2 2

Zz

log 03 (t + a)dt.

0

By taking logarithms on both sides of 1.4(3) and using the Maclaurin series of log(1 + x), we are led to the following identity (cf. Srivastava [1072, p. 13, Eq. (5.2)]): ∞ X ζ (k) k+1 z z2 (−1)k z = [1 − log(2π)] + (1 + γ ) + log G(z + 1) k+1 2 2

(|z| < 1).

k=2

(482)

312

Zeta and q-Zeta Functions and Associated Series and Integrals

Letting z → 1 in (482), in view of 1.3(6), we obtain ∞ X γ 1 ζ (k) = 1 + − log(2π), (−1)k k+1 2 2

(483)

k=2

which was proved by Suryanarayana [1138] and, again, by Singh and Verma [1033, p. 3, Section 4]. If we set z = 12 in (482) and make use of 1.4(6) and 1.3(8), we get ∞ X γ 5 ζ (k) −k 2 = 1+ − log 2 − 3 log A. (−1)k k+1 4 12

(484)

k=2

Multiplying each member of (5) through (8) by t, integrating both sides of the resulting equations from t = 0 to t = z and using (476), we obtain ∞ X k=2

ζ (k, a)

zk+1 ψ(a) 2 = z + z log 0(a − z) + k+1 2

Z−z log 0(a + t) dt 0

z2

z = [2a − 1 − log(2π)] + [ψ(a) − 1] + (a − 1) log 0(a − z) 2 2 − log G(a − z) + (1 − a) log 0(a) + log G(a) (|z| < |a|); Zz ∞ X zk+1 ψ(a) 2 k (−1) ζ (k, a) =− z + z log 0(a + z) − log 0(a + t) dt k+1 2 k=2

0

z2

z + [1 − ψ(a)] + (1 − a) log 0(a + z) 2 2 + log G(a + z) + (a − 1) log 0(a) − log G(a) (|z| < |a|),

= [2a − 1 − log(2π)]

(485)

(486)

which, in the special case when a = 1, corresponds to (482); Zz z2k+1 z 1 ζ (2k, a) = log 0(a + z)0(a − z) − log 0(a + t) dt 2k + 1 2 2 k=1 −z   1 0(a + z) G(a + z) = [2a − 1 − log(2π)]z + (1 − a) log + log 2 0(a − z) G(a − z)

∞ X

(|z| < |a|);

(487)

Series Involving Zeta Functions ∞ X

ζ (2k + 1, a)

k=1

313

z2k+2 k+1

0(a − z) + = ψ(a)z + z log 0(a + z) 2

Zz

Z−z log 0(a + t) dt + log 0(a + t) dt

0

(488)

0

= [ψ(a) − 1]z2 + (a − 1) log 0(a + z)0(a − z) − log G(a + z)G(a − z) + 2(1 − a) log 0(a) + 2 log G(a) (|z| < |a|). Taking a = 1 in (485), (487) and (488) and using 1.4(6), 2.3(2) and 1.3(4), we obtain ∞ X

z 1+γ 2 zk+1 = [1 − log(2π)] − z − log G(1 − z) (|z| < 1); k+1 2 2 k=2   ∞ X z2k+1 1 G(1 + z) ζ (2k) = [1 − log(2π)]z + log (|z| < 1); 2k + 1 2 G(1 − z) k=1 ∞ X

ζ (k)

ζ (2k + 1)

k=1

Setting z = ∞ X k=2 ∞ X k=1 ∞ X k=1

1 2

z2k+2 = −(1 + γ )z2 − log G(1 + z)G(1 − z) k+1

(|z| < 1).

(489)

(490)

(491)

in (489), (490) and (491) and using 1.4(6) and 1.4(8), we obtain

7 γ ζ (k) =− − log 2 + 3 log A; k 4 12 (k + 1) · 2

(492)

ζ (2k) 1 1 = − log 2; 2 2 (2k + 1) · 22k

(493)

ζ (2k + 1) 1 = −2 − γ − log 2 + 12 log A. 3 (k + 1) · 22k

(494)

Taking a = 2 in (485) through (488) and using 1.4(6), 2.3(2) and 1.3(50), we obtain ∞ X zk+1 z γ 0(2 − z) {ζ (k) − 1} = [3 − log(2π)] − z2 + log k+1 2 2 G(2 − z)

k=2 ∞ X

(−1)k {ζ (k) − 1}

k=2

(|z| < 2); (495)

z γ G(2 + z) zk+1 = [3 − log(2π)] + z2 + log k+1 2 2 0(2 + z)

(|z| < 2); (496)

314

Zeta and q-Zeta Functions and Associated Series and Integrals

  ∞ X z2k+1 1 G(2 + z)0(2 − z) {ζ (2k) − 1} = [3 − log(2π)]z + log 2k + 1 2 G(2 − z)0(2 + z)

(|z| < 2);

k=1

(497) ∞ X

z2k+2

k=1

k+1

{ζ (2k + 1) − 1}

= −γ z2 + log

0(2 + z)0(2 − z) G(2 + z)G(2 − z)

(|z| < 2).

(498)

Taking the limit as z → 2 on both sides of (496) and using 1.4(6), we obtain ∞ X 2k+1 (−1)k {ζ (k) − 1} = 3 + 2γ − log(6π). k+1

(499)

k=2

Setting z = 1, z = we obtain ∞ X ζ (k) − 1 k=2 ∞ X k=2 ∞ X k=1 ∞ X k=1 ∞ X k=2 ∞ X

k+1

k=1 ∞ X

and z =

3 2

in (495) through (498) and using 1.4(6) and 1.4(8),

3 γ 1 − − log(2π); 2 2 2

(501)

ζ (2k) − 1 3 1 = − log(4π); 2k + 1 2 2

(502)

ζ (2k + 1) − 1 = −γ + log 2; k+1

(503)

 31  ζ (k) − 1 5 γ 3 − 12 · A ; = − + log 2 4 4 (k + 1) · 2k

(504)

 19  ζ (k) − 1 7 γ = + + log 2 12 · 3−2 · A−3 ; k 4 4 (k + 1) · 2

 1  ζ (2k) − 1 3 −2 −1 = + log 2 · 3 ; 2 (2k + 1) · 22k

 25  ζ (2k + 1) − 1 −3 4 12 = −1 − γ + log 2 · 3 · A ; (k + 1) · 22k k=1   ∞   X ζ (k) − 1 3 k 17 3γ = − + log 2−frac1936 · A ; k+1 2 12 4 k=2   ∞  17  X 2 ζ (k) − 1 3 k 19 3γ (−1)k = + + log 2− 36 · 5− 3 · A−1 ; k+1 2 12 4 k=2

(500)

(−1)k 3 γ 1 {ζ (k) − 1} = + − log(8π); k+1 2 2 2

(−1)k

k=2 ∞ X

=

1 2

(505)

(506)

(507)

(508)

(509)

Series Involving Zeta Functions

  ∞ X ζ (2k) − 1 3 2k k=1 ∞ X k=1

2k + 1

2

ζ (2k + 1) − 1 k+1

Taking a =

1 2

315

=

  1 1 3 + log 2− 2 · 5− 3 ; 2

(510)

 2k  1 4 4 3 1 = − − γ + log 2− 27 · 5 9 · A 3 . 2 9

(511)

in (485) through (488) and using 1.4(8), 2.3(2) and 1.3(51), we obtain

∞   X zk+1 2k − 1 ζ (k) k+1 k=2

=

  1 z z2 1 1 − log(2π) − (1 + γ + 2 log 2) − log 0 −z 8 2 2 2 2     1 3 1 1 − log G − z + log 2 24 · A− 2 (|z| < ); 2 2

(512)

∞   X zk+1 (−1)k 2k − 1 ζ (k) k+1 k=2

  1 z z2 1 1 = − − log(2π) + (1 + γ + 2 log 2) + log 0 +z 8 2 2 2 2    1  3 1 1 + log G + z + log 2− 24 · A 2 (|z| < ); 2 2

(513)

∞   X z2k+1 22k − 1 ζ (2k) 2k + 1 k=1

   1 + z + z G 2 z 1 1  + log   = − log(2π) + log  1 2 4 2 0 2 −z G 12 − z 0



1 2

(514) 1 (|z| < ); 2

∞   X z2k+2 22k+1 − 1 ζ (2k + 1) k+1 k=1

    1 1 1 1 2 = − (1 + γ + 2 log 2)z − log 0 +z 0 −z 4 2 2 2      1  1 1 1 − log G +z G − z + log 2 12 · A−3 (|z| < ). 2 2 2 Taking the limit in (513) as z → ∞ X (−1)k k=2

1 2

(515)

and using 1.3(6), we obtain

 1  2k − 1 γ − 12 − 21 3 ζ (k) = + log 2 · π · A . 4 (k + 1) · 2k

(516)

316

Zeta and q-Zeta Functions and Associated Series and Integrals

Multiplying each member of (5) through (8) by t2 , integrating both sides of the resulting equations from t = 0 to t = z and using (478), we obtain ∞ X

ζ (k, a)

k=2

=

zk+2 k+2

1−a z3 z2 [1 − 2a + log(2π)] z + [1 − log(2π)] + [ψ(a) − 1] 2 4 3 (517) + (a − 1)2 log 0(a − z) − (z + a − 1) log G(a − z) − (a − 1)2 log 0(a) + (a − 1) log G(a) −

Z−z log G(t + a) dt

(|z| < |a|);

0 ∞ X zk+2 (−1)k ζ (k, a) k+2 k=2

=

z3 a−1 z2 [1 − 2a + log(2π)] z + [1 − log(2π)] + [1 − ψ(a)] 2 4 3 (518) + (a − 1)2 log 0(a + z) + (z − a + 1) log G(a + z) − (a − 1)2 log 0(a) + (a − 1) log G(a) −

Zz log G(t + a) dt

(|z| < |a|);

0 ∞ X

ζ (2k, a)

k=1

z2k+2 k+1

= 2(a − 1) log G(a) − 2(a − 1)2 log 0(a) + [1 − log(2π)]

z2 2

+ (a − 1)2 log 0(a + z)0(a − z) + (z − a + 1) log G(a + z) − (z + a − 1) log G(a − z) −

Zz log G(t + a) dt 0

Z−z log G(t + a) dt

−

(|z| < |a|);

0 ∞ X

ζ (2k + 1, a)

k=1

=

z2k+3 2k + 3

1−a z3 1 0(a + z) [1 − 2a + log(2π)]z + [ψ(a) − 1] − (a − 1)2 log 2 3 2 0(a − z)

(519)

Series Involving Zeta Functions

317

1 1 − (z − a + 1) log G(a + z) − (z + a − 1) log G(a − z) 2 2   z −z Z Z   1 (|z| < |a|). log G(t + a) dt − log G(t + a) dt +  2 0

(520)

0

If we use 1.4(6), 1.4(55), 1.4(56), 1.4(73) and 1.4(77), we readily find that Z−1 1 log 0(t + 1) dt = − log(2π); 2

(521)

0

Z−1 1 1 log G(t + 1) dt = − + log(2π) + 2 log A; 12 4

(522)

0 1

Z− 2

log 0(t + 1) dt = −

7 1 3 log 2 − log π + log A; 24 4 2

(523)

log G(t + 1) dt = −

1 1 1 1 7 + log 2 + log π + log A − log B. 24 12 16 4 4

(524)

0 1

Z− 2 0

Taking a = 1 in (517) through (520) and using 1.4(6), 2.3(2) and 1.3(4), we obtain ∞ X

ζ (k)

k=2

zk+2 k+2

= [1 − log(2π)]

z2 z3 3 − (1 + γ ) z log G(1 − z) 4 −

Z−z − log G(t + 1) dt

(525)

(|z| < 1);

0 ∞ X zk+2 (−1)k ζ (k) k+2 k=2

= [1 − log(2π)]

z2 z3 + (1 + γ ) + z log G(1 + z) 4 3

Zz log G(t + 1) dt

− 0

(|z| < 1);

(526)

318

Zeta and q-Zeta Functions and Associated Series and Integrals ∞ X

ζ (2k)

k=1

z2k+2 k+1

z2 G(1 + z) + z log 2 G(1 − z) z −z Z Z − log G(t + 1) dt − log G(t + 1) dt

= [1 − log(2π)]

0 ∞ X

(527) (|z| < 1);

0

ζ (2k + 1)

k=1

z2k+3 2k + 3

z3 1 = −(1 + γ ) − z log G(1 + z)G(1 − z) 2  z 3  Z Z−z   1 + log G(t + 1) dt − log G(t + 1) dt  2 0

(528) (|z| < 1).

0

Taking the limit in (526) as z → 1 and using 1.4(6) and 1.4(73), we obtain ∞   1 X 1 1 γ ζ (k) = + + log 2− 2 · π − 2 · A2 . (−1)k k+2 2 3

(529)

k=2

If we set z = we get ∞ X k=2 ∞ X

k=1

in (525) through (528), by virtue of 1.4(6), 1.4(8), 1.4(77) and (524),

 2  ζ (k) γ −3 2 7 + log 2 = − · A · B ; 6 (k + 2) · 2k

(−1)k

k=2 ∞ X

1 2

 1  1 γ ζ (k) −3 −2 7 = + + log 2 · A · B ; 2 6 (k + 2) · 2k

  1 ζ (2k) = + log 2−1 · B14 ; 2k 2 (k + 1) · 2

∞  1  X 1 γ ζ (2k + 1) −3 4 = − − + log 2 · A . 2 3 (2k + 3) · 22k

(530)

(531)

(532)

(533)

k=1

Now, we recall the following integral formula (cf. Choi et al. [308, p. 386]): Z1 log G(t + a) dt 0

=

a(1 − a) a + log(2π) + a log 0(a) − log G(a + 1) 2 2  1

1

1

+ log 2− 4 · π − 4 · e 12 · A−2 .

(534)

Series Involving Zeta Functions

319

In view of 1.4(6), (534) with a = 2 yields Z1 log G(t + 2) dt = −

11 3 + log(2π) − 2 log A. 12 4

(535)

0

By setting a = 2 in (517) through (520) and using 1.4(6), 2.3(2) and 1.3(50), we find that ∞ X zk+2 {ζ (k) − 1} k+2 k=2

z2 γ z + [1 − log(2π)] − z3 + log 0(2 − z) 2 4 3 Z−z − (z + 1) log G(2 − z) − log G(t + 2) dt (|z| < 2);

= [3 − log(2π)]

(536)

0 ∞ X zk+2 (−1)k {ζ (k) − 1} k+2 k=2

z z2 γ + [1 − log(2π)] + z3 + log 0(2 + z) 2 4 3 Zz + (z − 1) log G(2 + z) − log G(t + 2) dt (|z| < 2);

= [log(2π) − 3]

(537)

0 ∞ X z2k+2 {ζ (2k) − 1} k+1 k=1

z2 + log 0(2 + z)0(2 − z) + (z − 1) log G(2 + z) 2 Zz − (z + 1) log G(2 − z) − log G(t + 2) dt

= [1 − log(2π)]

0

Z−z − log G(t + 2) dt

(|z| < 2);

0 ∞ X z2k+3 {ζ (2k + 1) − 1} 2k + 3 k=1

= [3 − log(2π)]

z γ 3 1 0(2 + z) − z − log 2 3 2 0(2 − z)

(538)

320

Zeta and q-Zeta Functions and Associated Series and Integrals

1 1 − (z − 1) log G(2 + z) − (z + 1) log G(2 − z) 2 2  Zz Z−z   1 + (|z| < 2). log G(t + 2) dt − log G(t + 2) dt  2 0

(539)

0

Taking the limit in (537) as z → 2 and using 1.4(6), 1.4(55) and (535), we obtain ∞   1 X 1 2k 11 2γ (−1)k (ζ (k) − 1) = + + log 3 4 · π − 2 · A . k + 2 24 3

(540)

k=2

If use is made of 1.4(6), 1.4(56), 1.4(73), 1.4(77), (524) and (535), the following integral formulas are readily obtained: 3

Z2 log G(t + 2) dt 0

Z1

1

1

Z2

Z2 log G(t + 2) dt +

log G(t + 2) dt +

=

0

0

log 0(t + 2) dt

(541)

0

19 31 3 21 3 7 =− − log 2 + log 3 + log π + log A − log B; 8 24 2 16 4 4 3

Z− 2 log G(t + 2) dt 0 1

Z− 2

Z1

log G(t + 1) dt

log G(t + 1) dt +

=−

(542)

0

0

1 1 3 9 7 = − − log 2 − log π + log A − log B. 8 6 16 4 4 If we set z = 1, z = 12 , and z = 23 in (536) through (539), in view of 1.4(6), 1.4(8), 1.4(56), 1.4(73), 1.4(77), (523), (524), (535) and (541), we get ∞ X ζ (k) − 1 k=2 ∞ X

k+2

(−1)k

k=2

=

 1  1 ζ (k) − 1 13 γ =− + log 2 2 · π − 2 · A2 ; k+2 + 3

∞ X ζ (2k) − 1 k=1

 1  1 11 γ − + log 2− 2 · π − 2 · A−2 ; 6 3

k+1

=

3 − log π ; 2

(543)

(544)

(545)

Series Involving Zeta Functions ∞ X ζ (2k + 1) − 1 k=1 ∞ X k=2 ∞ X

2k + 3

k=1 ∞ X

  1 γ + log 2− 2 · A−2 ; 3

  14 ζ (k) − 1 8 γ 2 7 −3 · A · B ; = − + log 2 3 6 (k + 2) · 2k

(−1)k

k=2 ∞ X

= 1312 −

321

 13  ζ (k) − 1 7 γ −3 4 −2 7 = − + + log 2 · 3 · A · B ; 6 6 (k + 2) · 2k

  3 ζ (2k) − 1 −9 4 14 = + log 2 · 3 · B ; 2 (k + 1) · 22k

 1  ζ (2k + 1) − 1 23 γ 2 · 3−4 · A4 ; − + log 2 = 6 3 (2k + 3) · 22k k=1   ∞  5 2 7 X ζ (k) − 1 3 k 7 γ = − + log 2− 9 · A 3 · B 9 ; k+2 2 6 2 k=2   ∞  4 4  X 2 7 ζ (k) − 1 3 k 1 γ = + + log 2− 3 · 5 9 · A− 3 · B 9 ; (−1)k k+2 2 3 2 k=2   ∞  17 4 14  X ζ (2k) − 1 3 2k 3 = + log 2− 9 · 5 9 · B 9 ; k+1 2 2 k=1   ∞  7  X 4 4 ζ (2k + 1) − 1 3 2k 5 γ = − + log 2 27 · 5− 27 · A 9 . 2k + 3 2 18 3

(546)

(547)

(548)

(549)

(550)

(551)

(552)

(553)

(554)

k=1

Taking a =

1 2

in (517) through (520) and using 1.4(8), 2.3(2) and 1.3(51), we obtain

∞   X zk+2 2k − 1 ζ (k) k+2 k=2

 1  z 1 3 z2 = log 2− 48 · e− 16 · A 4 + log(2π) + [1 − log(2π)] 4 4       (555) 3 z 1 1 1 1 − (1 + γ + 2 log 2) + log 0 −z + − z log G −z 3 4 2 2 2     Z−z 1 1 − log G t + dt |z| < ; 2 2 0 ∞   X zk+2 (−1)k 2k − 1 ζ (k) k+2 k=2

 1  z 1 3 z2 = log 2− 48 · e− 16 · A 4 − log(2π) + [1 − log(2π)] 4 4

322

Zeta and q-Zeta Functions and Associated Series and Integrals

      z3 1 1 1 1 + (1 + γ + 2 log 2) + log 0 +z + + z log G +z 3 4 2 2 2     Zz 1 1 − log G t + dt |z| < ; (556) 2 2 0 ∞   X z2k+2 22k − 1 ζ (2k) k+1 k=1

      1 3 z2 1 1 1 − 24 − 18 · e · A 2 + [1 − log(2π)] + log 0 +z 0 −z = log 2 2 4 2 2 

       1 1 1 1 + z log G +z + − z log G −z 2 2 2 2 z −z     Z Z 1 1 1 − log G t + dt − log G t + dt (|z| < ); 2 2 2

(557)

+

0

0

∞   X z2k+3 22k+1 − 1 ζ (2k + 1) 2k + 3 k=1

=

z3



1 2

+z



1 z  log(2π) − (1 + γ + 2 log 2) − log  4 3 8 0 12 − z         1 1 1 1 1 1 + z log G +z + − z log G −z − 2 2 2 2 2 2  z  −z      Z Z   1  1 1 1 + dt − log G t + dt . log G t + |z| <  r2  2 2 2

By taking the limit in (556) as z → and 1.4(77), we obtain ∞ X (−1)k

(558)

0

0

k=2

0

1 2

and using 1.4(6), 1.4(55), 1.4(56), 1.4(73)

 23  1 γ 2k − 1 6 · π − 2 · A4 · B−7 . ζ (k) = + log 2 6 (k + 2) · 2k

(559)

Integrating both sides of (1) through (4) from t = 0 to t = z, in view of (470), we find that ∞ X k=2

ζ (k, a)

zk+1 k(k + 1)

z2 = −z log 0(a) + ψ(a) − 2

Z−z log 0(a + t) dt 0

Series Involving Zeta Functions

323

ψ(a) 2 = −z log 0(a) + z + 2

Zz

log 0(a − t) dt

0

 1 1 + log(2π) − a − log 0(a) z = (a − 1) log 0(a) − log G(a) + 2 2 

(560) z2 + [1 + ψ(a)] + (z − a + 1) log 0(a − z) + log G(a − z) (|z| < |a|); 2 Zz ∞ X zk+1 z2 k (−1) ζ (k, a) = −z log 0(a) − ψ(a) + log 0(a + t) dt k(k + 1) 2 k=2 0 (561)   1 1 = (1 − a) log 0(a) + log G(a) + + log(2π) − a − log 0(a) z 2 2 − [1 + ψ(a)] ∞ X k=1

ζ (2k, a)

z2 + (z + a − 1) log 0(z + a) − log G(z + a) 2

(|z| < |a|);

z2k+1 k(2k + 1)

= −2z log 0(a) +

Zz

−z Zz

= −2z log 0(a) +

log 0(a + t) dt (562) {log 0(a + t) + log 0(a − t)} dt

0

= [1 − 2a + log(2π) − 2 log 0(a)] z + (z + a − 1) log 0(a + z) + (z − a + 1) log 0(a − z) + log ∞ X

ζ (2k + 1, a)

k=1

= ψ(a)z − 2

(|z| < |a|);

z2k+2 (k + 1)(2k + 1)

Zz 0

= ψ(a)z2 +

G(a − z) G(a + z)

Zz

Z−z log 0(a + t) dt − log 0(a + t) dt 0

(563)

{log 0(a − t) − log 0(a + t)} dt

0

= 2(a − 1) log 0(a) − 2 log G(a) + [1 + ψ(a)]z2 + (1 − a + z) log 0(a − z) + (1 − a − z) log 0(a + z) + log G(a − z)G(a + z)

(|z| < |a|).

324

Zeta and q-Zeta Functions and Associated Series and Integrals

If we set a = 1 in (560) through (563), by virtue of 1.4(6), 2.3(2) and 1.3(4), we get ∞ X k=2

ζ (k)

zk+1 k(k + 1)

= [log(2π) − 1]

z z2 + (1 − γ ) + z log 0(1 − z) r2 2

+ log G(1 − z)

(564)

(|z| < 1);

∞ X zk+1 (−1)k ζ (k) k(k + 1) k=2

= [log(2π) − 1]

z2 z + (γ − 1) + z log 0(z + 1) 2 2

− log G(z + 1) ∞ X

ζ (2k)

k=1

(565)

(|z| < 1);

z2k+1 k(2k + 1)

= [log(2π) − 1] z + z log 0(1 − z)0(1 + z) + log ∞ X

G(1 − z) G(1 + z)

ζ (2k + 1)

k=1

(566)

(|z| < 1);

z2k+2 (k + 1)(2k + 1)

= (1 − γ )z2 + z log

0(1 − z) + log G(1 − z)G(1 + z) 0(1 + z)

(567) (|z| < 1).

Taking the limit in (565) as z → 1 and using 1.3(6), we obtain ∞ X γ 1 ζ (k) = −1 + + log(2π). (−1)k k(k + 1) 2 2

(568)

k=2

Setting z = ∞ X k=2

in (564) through (567) and using 1.4(6) and 1.4(8), we obtain

 7  1 ζ (k) γ 12 · π 2 · A−3 ; = − + log 2 4 k(k + 1) · 2k

∞ X (−1)k k=2

1 2

 7  1 γ ζ (k) − 12 2 · A3 ; = −1 + + log 2 · π 4 k(k + 1) · 2k

(569)

(570)

Series Involving Zeta Functions ∞ X k=1 ∞ X k=1

325

ζ (2k) = −1 + log π; k(2k + 1) · 22k

(571)

  7 ζ (2k + 1) 3 · A−12 . = 2 − γ + log 2 (k + 1)(2k + 1) · 22k

(572)

By putting a = 2 in (560) through (563) and using 1.4(6), 2.3(2) and 1.3(50), we obtain ∞ X zk+1 {ζ (k) − 1} k(k + 1) k=2

z2 z + (2 − γ ) + (z − 1) log 0(2 − z) 2 2 + log G(2 − z) (|z| < 2);

= [log(2π) − 3]

(573)

∞ X zk+1 (−1)k {ζ (k) − 1} k(k + 1) k=2

z2 z + (γ − 2) + (z + 1) log 0(z + 2) 2 2 − log G(z + 2) (|z| < 2);

= [log(2π) − 3]

(574)

∞ X z2k+1 {ζ (2k) − 1} k(2k + 1) k=1

= [log(2π) − 3] z + (z + 1) log 0(2 + z) + (z − 1) log 0(2 − z) G(2 − z) (|z| < 2); + log G(2 + z) ∞ X z2k+2 {ζ (2k + 1) − 1} (k + 1)(2k + 1) k=1

= (2 − γ )z2 + (z − 1) log 0(2 − z) − (z + 1) log 0(2 + z) + log G(2 − z)G(2 + z) (|z| < 2).

(575)

(576)

Taking the limit in (574) as z → 2 and using 1.4(6), we have ∞  3 3 1 X {ζ (k) − 1}2k 7 (−1)k = − + γ + log 2 2 · 3 2 · π 2 . k(k + 1) 2

(577)

k=2

If we set z = 1, z = we obtain ∞ X ζ (k) − 1 k=2

1 2

and z =

3 2

in (573) through (576), in view of 1.4(6) and 1.4(8),

 1 1 1 γ = − − + log 2 2 · π 2 ; k(k + 1) 2 2

(578)

326

Zeta and q-Zeta Functions and Associated Series and Integrals ∞  5 1 X ζ (k) − 1 5 γ (−1)k = − + + log 2 2 · π 2 ; k(k + 1) 2 2

(579)

k=2

∞ X ζ (2k) − 1 k=1

k(2k + 1)

= −3 + log(8π);

(580)

∞ X ζ (2k + 1) − 1 = 2 − γ − 2 log 2; (k + 1)(2k + 1)

(581)

∞  19 1  X ζ (k) − 1 3 γ 12 · π 2 · A−3 ; = − − + log 2 4 4 k(k + 1) · 2k

(582)

∞  43  X 1 ζ (k) − 1 9 γ − 12 3 2 · A3 ; (−1)k = − + + log 2 · 3 · π 4 4 k(k + 1) · 2k

(583)

k=1

k=2

k=2

∞ X k=1 ∞ X k=1

  ζ (2k) − 1 −2 3 = −3 + log 2 · 3 · π ; k(2k + 1) · 22k

(584)

 31  ζ (2k + 1) − 1 = 3 − γ + log 2 3 · 3−6 · A−12 ; 2k (k + 1)(2k + 1) · 2

(585)

  ∞ X ζ (k) − 1 3 k k=2

k(k + 1)

2

=

 19 1  1 3γ − + log 2 36 · π 2 · A−1 ; 12 4

 k ∞  91  X 5 1 37 3γ k ζ (k) − 1 3 (−1) =− + + log 2− 36 · 3 · 5 3 · π 2 · A ; k(k + 1) 2 12 4

(586)

(587)

k=2

  ∞ X ζ (2k) − 1 3 2k k=1

k(2k + 1)

2

  5 = −3 + log 2−2 · 3 · 5 3 · π ;

  ∞  55  X 2 10 4 ζ (2k + 1) − 1 3 2k 19 = − γ + log 2 27 · 3− 3 · 5− 9 · A− 3 . (k + 1)(2k + 1) 2 9

(588)

(589)

k=1

If we take a = obtain

1 2

in (560) through (563), by virtue 1.4(8), 2.3(2) and 1.3(51), we

∞   X zk+1 2k − 1 ζ (k) k(k + 1) k=2

 1  z 1 3 z2 = log 2− 24 · e− 8 · A 2 + log 2 + (1 − γ − 2 log 2) 2       2  1 1 1 1 + + z log 0 − z + log G −z |z| < ; 2 2 2 2

(590)

Series Involving Zeta Functions

327

∞   X zk+1 (−1)k 2k − 1 ζ (k) k(k + 1) k=2

 1 1  z 3 z2 = log 2 24 · e 8 · A− 2 + log 2 + (γ + 2 log 2 − 1)    2    2  1 1 1 1 log 0 z + − log G z + |z| < ; + z− 2 2 2 2 ∞   X z2k+1 22k − 1 ζ (2k) k(2k + 1) k=1         1 1 1 1 = z log 2 + z − log 0 +z + + z log 0 −z 2 2 2 2     G 12 − z 1  + log  ; |z| < 2 G 1 +z

(591)

(592)

2

∞  X



22k+1 − 1 ζ (2k + 1)

k=1

z2k+2 (k + 1)(2k + 1)

  1 1 = log 2− 12 · e− 4 · A3 + (1 − γ − 2 log 2)z2         1 1 1 1 + + z log 0 −z + − z log 0 +z 2 2 2 2       1 1 1 −z G +z |z| < . + log G 2 2 2 Taking the limit in (591) as z → ∞ X (−1)k k=2

1 2

(593)

and using 1.3(6), we obtain

 13  2k − 1 γ 12 · A−3 . ζ (k) = + log 2 4 k(k + 1) · 2k

(594)

By setting z = t in (560) through (563), integrating both sides of the resulting equations from t = 0 to t = z and using (470) and (478), we obtain ∞ X

ζ (k, a)k(k + 1)(k + 2)zk+2

k=2

1 1 = (a − 1) log G(a) − (a − 1)2 log 0(a) 2 2  1 + (a − 1)[2a − 1 − log(2π)] + (a − 1) log 0(a) − log G(a) z 4 + [5 − 8a + 3 log(2π) − 4 log 0(a)]

z2 z3 + [2 + ψ(a)] 8 6

(595)

328

Zeta and q-Zeta Functions and Associated Series and Integrals

1 1 + (z − a + 1)2 log 0(a − z) + (z − a + 1) log G(a − z) 2 2 Z−z 1 − log G(t + a) dt (|z| < |a|); 2 0 ∞ X (−1)k k=2

ζ (k, a) zk+2 k(k + 1)(k + 2)

1 1 = (a − 1) log G(a) − (a − 1)2 log 0(a) 2 2  1 (a − 1)[1 − 2a + log(2π)] + (1 − a) log 0(a) + log G(a) z + 4 z2 z3 + [5 − 8a + 3 log(2π) − 4 log 0(a)] − [2 + ψ(a)] 8 6 1 1 2 + (z + a − 1) log 0(z + a) − (z + a − 1) log G(z + a) 2 2 Zz 1 log G(t + a) dt (|z| < |a|); − 2

(596)

0 ∞ X k=1

ζ (2k, a) z2k+2 = 2(a − 1) log G(a) k(k + 1)(2k + 1) − 2(a − 1)2 log 0(a) + [5 − 8a + 3 log(2π) − 4 log 0(a)]

z2 2

+ (z + a − 1)2 log 0(a + z) + (z − a + 1)2 log 0(a − z) + (1 − a − z) log G(a + z) + (1 − a + z) log G(a − z) Zz −

Z−z log G(t + a) dt − log G(t + a) dt

0

(597)

(|z| < |a|);

0

∞ X

ζ (2k + 1, a) z2k+3 (k + 1)(2k + 1)(2k + 3) k=1   1 = 2(a − 1) log 0(a) − 2 log G(a) + (a − 1)[2a − 1 − log(2π)] z 2 z3 1 1 + (z − a + 1)2 log 0(a − z) − (z + a − 1)2 log 0(a + z) 3 2 2 1 1 + (z − a + 1) log G(a − z) + (z + a − 1) log G(a + z) 2 2 Zz Z−z 1 1 + log G(t + a) dt − log G(t + a) dt (|z| < |a|). 2 2 + [2 + ψ(a)]

0

0

(598)

Series Involving Zeta Functions

329

Taking a = 1 in (595) through (598) and using 1.4(6), 2.3(2) and 1.3(4), we obtain ∞ X k=2

ζ (k) zk+2 k(k + 1)(k + 2) 3z2 z3 z2 + (2 − γ ) + log 0(1 − z) 8 6 2 Z−z 1 z + log G(1 − z) − log G(t + 1) dt (|z| < 1); 2 2

= [log(2π) − 1]

(599)

0 ∞ X (−1)k

ζ (k) zk+2 k(k + 1)(k + 2)

k=2

3z2 z3 z2 + (γ − 2) + log 0(z + 1) 8 6 2 Zz z 1 − log G(z + 1) − log G(t + 1) dt (|z| < 1); 2 2

= [log(2π) − 1]

(600)

0 ∞ X k=1

ζ (2k) z2k+2 k(k + 1)(2k + 1)

= [log(2π) − 1] Zz −

Z−z log G(t + 1) dt − log G(t + 1) dt

0 ∞ X k=1

3z2 G(1 − z) + z2 log 0(1 + z)0(1 − z) + z log 2 G(1 + z)

(601)

(|z| < 1);

0

ζ (2k + 1) z2k+3 (k + 1)(2k + 1)(2k + 3)

= (2 − γ )

+

1 2

z3 z2 0(1 − z) z + log + log G(1 − z)G(1 + z) 3 2 0(1 + z) 2

Zz log G(t + 1) dt − 0

1 2

Z−z log G(t + 1) dt

(602)

(|z| < 1).

0

Taking the limit in (600) as z → 1 and using 1.4(6) and 1.4(73), we obtain ∞ X (−1)k k=2

 1 1  ζ (k) 3 γ = − + + log 2 4 · π 4 · A . k(k + 1)(k + 2) 4 6

(603)

330

Zeta and q-Zeta Functions and Associated Series and Integrals 1 2

If we set z = we get ∞ X k=2

  1 1 7 ζ (k) γ 4 · π 4 · A−2 · B 2 ; = − + log 2 12 k(k + 1)(k + 2) · 2k

∞ X (−1)k k=2 ∞ X k=1 ∞ X k=1

in (599) through (602), in view of 1.4(6), 1.4(8), 1.4(77) and (524),

 1 1  7 ζ (k) 3 γ −4 4 · A2 · B 2 ; = − + + log 2 · π 4 12 k(k + 1)(k + 2) · 2k

(604)

(605)

  3 ζ (2k) 14 + log π · B ; = − 2 k(k + 1)(2k + 1) · 22k

(606)

  ζ (2k + 1) 3 γ −8 − + log 2 · A . = 2 3 (k + 1)(2k + 1)(2k + 3) · 22k

(607)

Taking a = 2 in (595) through (598) and using 1.4(6), 2.3(2) and 1.3(50), we have ∞ X k=2

ζ (k) − 1 zk+2 k(k + 1)(k + 2)

= [3 − log(2π)]z4 + [3 log(2π) − 11]z2 8 + (3 − γ )z3 6

(608)

+ 12(z − 1)2 log 0(2 − z) + 12(z − 1) log G(2 − z) Z−z − 12 log G(t + 2) dt

(|z| < 2);

0 ∞ X

(−1)k

k=2

ζ (k) − 1 zk+2 k(k + 1)(k + 2)

z z2 z3 = [log(2π) − 3] + [3 log(2π) − 11] + (γ − 3) 4 8 6 1 1 + (z + 1)2 log 0(z + 2) − (z + 1) log G(z + 2) 2 2 Zz 1 − log G(t + 2) dt (|z| < 2); 2

(609)

0 ∞ X k=1

ζ (2k) − 1 z2k+2 k(k + 1)(2k + 1)

z2 = [3 log(2π) − 11] + (z + 1)2 log 0(2 + z) 2

(610)

Series Involving Zeta Functions

331

+ (z − 1)2 log 0(2 − z) − (z + 1) log G(z + 2) + (z − 1) log G(2 − z) Zz −

Z−z log G(t + 2) dt − log G(t + 2) dt

0 ∞ X k=1

(|z| < 2);

0

ζ (2k + 1) − 1 z2k+3 (k + 1)(2k + 1)(2k + 3)

z z3 1 = [3 − log(2π)] + (3 − γ ) + (z − 1)2 log 0(2 − z) 2 3 2 (611) 1 1 1 2 − (z + 1) log 0(2 + z) + (z − 1) log G(2 − z) + (z + 1) log G(2 + z) 2 r2 2 Zz Z−z 1 1 + log G(t + 2) dt − log G(t + 2) dt (|z| < 2). 2 2 0

0

Taking the limit in (609) as z → 2 and using 1.4(6) and (535), we obtain ∞ X (−1)k k=2

 3 9 1 1 ζ (k) − 1 109 γ 2k = − + + log 2 4 · 3 8 · π 4 · A 2 . k(k + 1)(k + 2) 48 3

(612)

If we set z = 1, z = 12 and z = 23 in (608) through (611), by virtue of 1.4(6), 1.4(8), 1.4(73), (523), (524), (535), (541) and (542), we get ∞ X k=2 ∞ X

  1 1 ζ (k) − 1 1 γ = − − + log 2 4 · π 4 · A−1 ; k(k + 1)(k + 2) 12 6

(−1)k

k=2 ∞ X k=1 ∞ X k=1 ∞ X k=2 ∞ X

(614)

9 ζ (2k) − 1 = − + 5 log 2 + log π; k(k + 1)(2k + 1) 2

(615)

  ζ (2k + 1) − 1 25 γ = − + log 2−2 · A−2 ; (k + 1)(2k + 1)(2k + 3) 12 3

(616)

 1 1  7 ζ (k) − 1 1 γ −4 4 · A−2 · B 2 ; = − + log 2 · π 3 12 k(k + 1)(k + 2) · 2k

(617)

(−1)k

k=2

 9 1  ζ (k) − 1 13 γ = − + + log 2 4 · π 4 · A ; k(k + 1)(k + 2) 6 6

(613)

 19 9 1  7 ζ (k) − 1 31 γ −4 2 · π 4 · A2 · B 2 ; = − + + log 2 · 3 12 12 k(k + 1)(k + 2) · 2k (618)

∞ X k=1

ζ (2k) − 1 k(k + 1)(2k + 1) · 22k

 9 = − + log 2−10 · 39 · π · B14 ; 2 

(619)

332

Zeta and q-Zeta Functions and Associated Series and Integrals ∞ X

  59 ζ (2k + 1) − 1 35 γ 6 · 3−9 · A−8 ; (620) = − + log 2 6 3 (k + 1)(2k + 1)(2k + 3) · 22k k=1  k ∞   1 1 X 2 7 ζ (k) − 1 3 1 γ (621) = − − + log 2 4 · π 4 · A− 3 · B 18 ; k(k + 1)(k + 2) 2 12 4 k=2  k ∞ X ζ (k) − 1 3 k (−1) k(k + 1)(k + 2) 2 k=2 (622)  61 1 25 1 2  7 13 γ = − + + log 2− 36 · 3 2 · 5 18 · π 4 · A 3 · · B 18 ; 6 4  2k ∞  26  X 25 14 ζ (2k) − 1 3 9 = − + log 2− 9 · 3 · 5 9 · π · B 9 ; (623) k(k + 1)(2k + 1) 2 2 k=1  2k ∞  35  X 1 25 8 25 γ ζ (2k + 1) − 1 3 = − + log 2 27 · 3− 3 · 5− 27 · A− 9 . (k + 1)(2k + 1)(2k + 3) 2 18 3 k=1

(624)

Taking a =

1 2

in (595) through (598) and using 1.4(6), 2.3(2) and 1.3(51), we get

∞  1  X 1 3 (2k − 1)ζ (k) k+2 z = log 2− 96 · e− 32 · A 8 k(k + 1)(k + 2) k=2   z2 1 3 1 1 z + [1 + 3 log 2 + log π ] log 2 + log π + log A − + 12 8 2 8 8   2  3 z 1 1 1 + [2 − γ − 2 log 2] + −z z+ log 0 6 2 2 2

1 + 2



     Z−z 1 1 1 1 + z log G −z − log G t + dt 2 2 2 2

(625)

  1 |z| < ; 2

0 ∞  1  X 1 3 (2k − 1)ζ (k) k+2 (−1)k z = log 2− 96 · e− 32 · A 8 k(k + 1)(k + 2) k=2   1 1 1 3 z2 + − log 2 − log π − log A z + [1 + 3 log 2 + log π ] 8 12 8 2 r8  2   3 z 1 1 1 + [γ − 2 + 2 log 2] + z− log 0 z + 6 2 2 2 z         Z 1 1 1 1 1 1 log G t + + − z log G +z − dt |z| < ; 2 2 2 2 2 2 0

(626)

Series Involving Zeta Functions

333

∞ X (22k − 1)ζ (2k) 2k+2 z k(k + 1)(2k + 1) k=1

 1  1 3 z2 = log 2− 24 · e− 8 · A 2 + [1 + 3 log 2 + log π ] 2  2    2   1 1 1 1 + − z log 0 +z + + z log 0 −z 2 2 2 2         1 1 1 1 − z log G +z + + z log G −z + 2 2 2 2 Zz



1 log G t + 2

−



  Z−z 1 dt − log G t + dt 2

0 ∞ X



(627)

 1 |z| < ; 2

0

(22k+1 − 1)ζ (2k + 1)

z2k+3 (k + 1)(2k + 1)(2k + 3) k=1   1 1 z3 1 = log 2 + log π + 3 log A − z + [2 − γ − 2 log 2] 6 4 4 3  2    2   1 1 1 1 1 1 + + z log 0 −z − − z log 0 +z 2 2 2 2 2 2         1 1 1 1 1 1 + z log G −z + z− log G z + + 2 2 2 2 2 2 1 + 2

Zz



1 log G t + 2



1 dt − 2

0

  Z−z 1 dt log G t + 2

(628)

  1 |z| < . 2

0

By using 1.4(6), 1.4(55), 1.4(56), 1.4(73) and 1.4(77), we can show that 1

Z2



1 log G t + 2



1 1 dt = − + log 2 2 2

0

Z1 +

log G (t + 1) dt −

0

Z1

(629)

0 1

1

Z2

Z2

− 0

log 0 (t + 1) dt

log G (t + 1) dt +

log 0 (t + 1) dt

0

1 13 1 1 7 = − log 2 − log π − log A + log B. 24 12 16 4 4

334

Zeta and q-Zeta Functions and Associated Series and Integrals

Taking the limit in (626) as z → ∞ X (−1)k k=2

1 2

and applying 1.4(6) and (629), we obtain

 5  7 (2k − 1)ζ (k) γ 2 · A−1 · B− 2 . = + log 2 12 k(k + 1)(k + 2) · 2k

(630)

By setting z = t in (485) through (488), integrating both sides of the resulting equations from t = 0 to t = z and using (470), we obtain ∞ X

ζ (k, a) zk+2 = (1 − a)2 log 0(a) + (1 − a) log G(a) (k + 1)(k + 2) k=2   1 (a − 1)[1 + log(2π)] + a(1 − a) + (1 − a) log 0(a) + log G(a) z + 2 z2 z3 + [ψ(a) − 1] 4 6 + (a − 1)(z − a + 1) log 0(a − z) + (a − 1) log G(a − z)

+ [4a − 3 − log(2π)]

+

Z−z log G(t + a) dt

(631)

(|z| < |a|);

0 ∞ X

(−1)k

k=2

 +

ζ (k, a) zk+2 = (1 − a)2 log 0(a) + (1 − a) log G(a) (k + 1)(k + 2)

 1 (1 − a)[1 + log(2π)] + a(a − 1) + (a − 1) log 0(a) − log G(a) z 2

z2 z3 + [1 − ψ(a)] 4 6 + (1 − a)(z + a − 1) log 0(z + a) + (a − 1) log G(z + a)

+ [4a − 3 − log(2π)]

Zz log G(t + a) dt

+

(632)

(|z| < |a|);

0 ∞ X k=1

ζ (2k, a) z2k+2 = 2(1 − a)2 log 0(a) + 2(1 − a) log G(a) (k + 1)(2k + 1)

z2 + (1 − a)(z + a − 1) log 0(a + z) 2 + (a − 1)(z − a + 1) log 0(a − z) + (a − 1) log G(a + z)G(a − z) + [4a − 3 − log(2π)] Zz + 0

Z−z log G(t + a) dt + log G(t + a) dt 0

(|z| < |a|);

(633)

Series Involving Zeta Functions

335

∞ X ζ (2k + 1, a) 2k+3 z (k + 1)(2k + 3) k=1

= {(a − 1)[1 + log(2π)] + 2a(1 − a) + 2(1 − a) log 0(a) + 2 log G(a)} z z3 + (a − 1)(z + a − 1) log 0(a + z) 3 G(a − z) + (a − 1)(z − a + 1) log 0(a − z) + (a − 1) log G(a + z)

+ [ψ(a) − 1]

Zz −

Z−z log G(t + a) dt + log G(t + a) dt

0

(634)

(|z| < |a|).

0

Setting a = 1 in (631) through (634) and using 1.4(6), 2.3(2) and 1.3(4), we obtain ∞ X k=2

ζ (k) zk+2 (k + 1)(k + 2)

z2 z3 = [1 − log(2π)] − (1 + γ ) + 4 6

Z−z log G(t + 1) dt

(635) (|z| < 1);

0 ∞ X (−1)k k=2

ζ (k) zk+2 (k + 1)(k + 2)

z2 z3 = [1 − log(2π)] + (1 + γ ) + 4 6

(636)

Zz log G(t + 1) dt

(|z| < 1);

0 ∞ X k=1

ζ (2k) z2k+2 (k + 1)(2k + 1)

= [1 − log(2π)]

z2 2

Zz +

Z−z log G(t + 1) dt + log G(t + 1) dt

0 ∞ X k=1

(637) (|z| < 1);

0

ζ (2k + 1) z2k+3 (k + 1)(2k + 3)

z3 = −(1 + γ ) − 3

Zz 0

Z−z log G(t + 1) dt + log G(t + 1) dt 0

(638) (|z| < 1);

336

Zeta and q-Zeta Functions and Associated Series and Integrals

Taking the limit in (636) as z → 1 and using 1.4(6) and 1.4(73), we obtain ∞ X (−1)k k=2

Setting z = we obtain ∞ X k=2

k=2

k=1 ∞ X k=1

1 2

(639)

in (635) through (638), and using 1.4(6), 1.4(8), 1.4(77), and (524),

 1  ζ (k) γ 12 · A · B−7 ; + log 2 = − 12 (k + 1)(k + 2) · 2k

∞ X (−1)k ∞ X

ζ (k) 1 γ = + − 2 log A. (k + 1)(k + 2) 2 6

 1  ζ (k) 1 γ − 12 −1 −7 = + + log 2 · A · B ; 2 12 (k + 1)(k + 2) · 2k

(640)

(641)

ζ (2k) 1 = − 14 log B; 2k 2 (k + 1)(2k + 1) · 2

(642)

 1  γ ζ (2k + 1) 3 · A4 . = −1 − + log 2 3 (k + 1)(2k + 3) · 22k

(643)

If we take a = 2 in (631) through (634), in view of 1.4(6), 2.3(2) and 1.3(50), we get ∞ X k=2

ζ (k) − 1 zk+2 (k + 1)(k + 2)

z z2 γ = [log(2π) − 3] + [5 − log(2π)] − z3 + (z − 1) log 0(2 − z) 2 4 6 −z Z + log G(2 − z) + log G(t + 2) dt (|z| < 2);

(644)

0 ∞ X (−1)k k=2

ζ (k) − 1 zk+2 (k + 1)(k + 2)

z z2 γ = [3 − log(2π)] + [5 − log(2π)] + z3 − (z + 1) log 0(z + 2) 2 4 6 Zz + log G(z + 2) + log G(t + 2) dt (|z| < 2); 0

(645)

Series Involving Zeta Functions ∞ X

337

ζ (2k) − 1(k + 1)(2k + 1)z2k+2

k=1

z2 − (z + 1) log 0(2 + z) 2 + (z − 1) log 0(2 − z) + log G(2 + z)G(2 − z)

= [5 − log(2π)] Zz

log G(t + 2) dt +

+ 0

Z−z log G(t + 2) dt

(646)

(|z| < 2);

0

∞ X

ζ (2k + 1) − 1 2k+3 z (k + 1)(2k + 3) k=1 γ = [log(2π) − 3]z − z3 + (z − 1) log 0(2 − z) + (z + 1) log 0(2 + z) 3 Zz Z−z G(2 − z) + log − log G(t + 2) dt + log G(t + 2) dt (|z| < 2). G(2 + z) 0

(647)

0

Taking the limit in (645) as z → 2 and using 1.4(6), 1.4(73) and (535), we obtain ∞   3 X 25 γ (ζ (k) − 1)2k = + + log 3− 4 · A−1 . (−1)k (k + 1)(k + 2) 24 3

(648)

k=2

Setting z = 21 , z = 1 and z = 23 in (644) through (647) and using 1.4(6), 1.4(8), 1.4(73), 1.4(77), (524), (535) and (541), we get ∞ X k=2 ∞ X

 25  ζ (k) − 1 17 γ 12 · A · B−7 ; = − − + log 2 12 12 (k + 1)(k + 2) · 2k

(−1)k

k=2 ∞ X k=1 ∞ X k=1 ∞ X k=2

(650)

  ζ (2k) − 1 3 8 −6 −14 = + log 2 · 3 · B ; 2 (k + 1)(2k + 1) · 22k

(651)

 28  ζ (2k + 1) − 1 26 γ −3 12 4 2 = − − + log · 3 · A ; 3 3 (k + 1)(2k + 3) · 22k

(652)

ζ (k) − 1 1 γ = − − + 2 log A; (k + 1)(k + 2) 3 6

(653)

∞ X (−1)k k=2

 71  ζ (k) − 1 35 γ = + + log 2 12 · 3−6 · A−1 · B−7 ; k 12 12 (k + 1)(k + 2) · 2

(649)

ζ (k) − 1 11 γ = + − 2 log (2A); (k + 1)(k + 2) 6 6

(654)

338

Zeta and q-Zeta Functions and Associated Series and Integrals ∞ X k=1

ζ (2k) − 1 3 = − 2 log 2; (k + 1)(2k + 1) 2

(655)

∞   X 13 γ ζ (2k + 1) − 1 = − − + log 22 · A4 ; (k + 1)(2k + 3) 6 3

(656)

k=1

∞ X k=2

ζ (k) − 1 (k + 1)(k + 2)

∞ X (−1)k k=2 ∞ X k=1

 k  1  7 1 3 1 γ = − + log 2 36 · A 3 · B− 9 ; 2 4 4

ζ (k) − 1 (k + 1)(k + 2)

ζ (2k) − 1 (k + 1)(2k + 1)

 k  31  10 1 7 3 5 γ = + + log 2 36 · 5− 9 · A− 3 · B− 9 ; 2 4 4

 2k  8  10 14 3 3 = + log 2 9 · 5− 9 · B− 9 ; 2 2

  ∞  5 20 4  X ζ (2k + 1) − 1 3 2k 2 γ = − − + log 2− 9 · 5 27 · A 9 . (k + 1)(2k + 3) 2 3 3

(657)

(658)

(659)

(660)

k=1

Taking a =

1 2

in (631) through (634) and using 1.4(8), 2.3(2) and 1.3(51), we obtain

∞ X (2k − 1)ζ (k) k+2 z (k + 1)(k + 2) k=2   1  5 3 1 1 1 z = log 2 48 · e 16 · A− 4 − log 2 + log π − + 3 log A 12 2 4 2    (661)  z2 z3 1 1 1 − [1 + log(2π)] − (1 + γ + 2 log 2) − + z log 0 −z 4 6 2 2 2

  Z−z   1 1 1 − log G − z + log G t + dt 2 2 2



|z| <

 1 ; 2

0 ∞ X (2k − 1)ζ (k) k+2 (−1)k z (k + 1)(k + 2) k=2   1  5 1 1 1 z − 34 48 16 = log 2 · e · A + log 2 + log π − + 3 log A 12 2 4 2     z2 z3 1 1 1 − [1 + log(2π)] + (1 + γ + 2 log 2) + z − log 0 z + 4 6 2 2 2 z   Z     1 1 1 1 − log G z + + log G t + dt |z| < ; 2 2 2 2 0

(662)

Series Involving Zeta Functions

339

∞ X (22k − 1)ζ (2k) 2k+2 z (k + 1)(2k + 1) k=1

     1 1  3 z2 1 1 1 = log 2 24 · e 8 · A− 2 − [1 + log(2π)] + z− log 0 z + 2 2 2 2         (663) 1 1 1 1 1 1 − + z log 0 − z − log G +z G −z 2 2 2 2 2 2 Zz

  Z−z 1 dt log G (t + 12) dt + log G t + 2

0

0

+

∞ X (22k+1 − 1)ζ (2k + 1) k=1

(k + 1)(2k + 3) 

=−

  1 |z| < ; 2

z2k+3

 5 1 1 z3 log 2 + log π − + 3 log A z − (1 + γ + 2 log 2) 12 2 4 3

  1        G + z 2 1 1 1 1 1 1 1  − z log 0 +z − + z log 0 − z + log  + 1 2 2 2 2 2 2 2 G −z 

2

Zz −



1 log G t + 2



  Z−z 1 dt + log G t + dt 2

Taking the limit in (662) as z →

k=2

 1 |z| < . 2

(664)

0

0

∞ X (−1)k



1 2

and using 1.4(6) and (629), we obtain

 47  γ (2k − 1)ζ (k) − 12 −1 7 = + log 2 · A · B . 12 (k + 1)(k + 2) · 2k

(665)

Evaluation in Terms of Catalan’s Constant G We now evaluate  series associated with the Zeta function in terms of Catalan’s constant G and 0 41 , by using some of the formulas we have already developed in this section. Setting z = 41 and z = 34 in (482) and using 1.3(6), (1.12), 1.3(23) and 1.3(24), we obtain ∞ X (−1)k

   ζ (k) G 1 − 12 − 21 − 29 = 1 + γ 8 − + log 2 · π · A · 0 ; (666) π 4 (k + 1) · 22k k=2 "  k   −1 # ∞ X 1 3 3 3γ G 1 k ζ (k) − (−1) = 1+ + + log π 2 · A 2 · 0 . (667) k+1 4 8 3π 4 k=2

340

Zeta and q-Zeta Functions and Associated Series and Integrals

Setting z = 14 and z = and 1.4(24), we obtain

3 4

in (489), (490) and (491) and using 1.4(6), 1.1(12), 1.4(23)

∞ X

"   −1 # 1 9 ζ (k) γ G 1 = − − + log π 2 · A 2 · 0 ; 8 π 4 (k + 1) · 22k k=2      ∞ X 3 3γ G 1 ζ (k) 3 k − 12 − 12 =− + + log 2 · π · A 2 · 0 ; rk + 1 4 8 3π 4 k=2 ∞ X

ζ (2k) 1 G 1 = − − log 2; 2 π 4 (2k + 1) · 24k k=1   ∞ X ζ (2k) 3 2k 1 G 1 = + − log 2; 2k + 1 4 2 3π 4 k=1 "   −8 # ∞ X ζ (2k + 1) 1 2 4 36 = −4 − γ + log 2 · π · A · 0 ; 4k 4 (k + 1) · 2 k=1 "      8 # ∞ X 3 ζ (2k + 1) 3 2k 1 − 23 − 43 4 = −43 − γ + log 2 · π · A · 0 . k+1 4 4

(668)

(669)

(670)

(671)

(672)

(673)

k=1

If we set z = 14 , z = 34 , z = 1.4(23) and 1.4(24), we get

5 4

and z =

7 4

in (495) to (498), in view of 1.4(6), 1.1(12),

   ∞ X 1 9 15 15 γ G 1 k ζ (k) − 1 −4 − − (−1) = + − + log 2 2 · 5 · π 2 · A 2 · 0 ; 2k 8 8 π 4 (k + 1) · 2 k=2

(674)  k ∞ X k ζ (k) − 1 3 (−1) k+1 4 k=2

(675) "   −1 # 8 4 1 3 G 1 13 3γ + + + log 2 3 · 7− 3 · π 2 · A− 2 · 0 ; = 8 8 3π 4 "   −1 # ∞ X 1 9 9 γ G ζ (k) − 1 1 −8 4 = − − + log 2 · 3 · π 2 · A 2 · 0 ; (676) 2k 8 8 π 4 (k + 1) · 2 k=2      ∞ X 3 ζ (k) − 1 3 k 3γ G 1 − 19 − 12 6 2 = 118 − + + log 2 ·π ·A ·0 ; (677) k+1 4 8 3π 4 k=2

∞ h 1 i X ζ (2k) − 1 3 G −4 2 −2 = − + log 2 · 3 · 5 ; 2 π (2k + 1) · 24k k=1   ∞ h 1 i X 2 ζ (2k) − 1 3 2k 3 G = + + log 2− 4 · 7− 3 ; 2k + 1 4 2 3π k=1

(678)

(679)

Series Involving Zeta Functions ∞ X ζ (2k + 1) − 1 k=1

(k + 1) · 24k

341

" = −3 − γ + log 2

−62

  −8 # 1 ·3 ·5 ·π ·A · 0 ; 4 16

16

4

36

(680)    8 #   ∞ X 3 16 4 1 1 ζ (2k + 1) − 1 3 2k − 4 − 70 ; = − − γ + log 2 9 · 7 9 · π 3 · A · 0 k+1 4 3 4 "

k=1

(681)   ∞ X ζ (k) − 1 5 k (−1)k k+1 4 k=2

   1 8 9 1 63 5γ 1 G ; + − + log 2− 2 · π − 2 · 3− 5 · A− 10 · 0 40 8 5π 4   ∞ X ζ (k) − 1 7 k (−1)k k+1 4 k=2 "   −1 # 9 4 1 G 87 7γ 1 − 14 − 47 7 2 + + + log 3 · 11 · π · A · 0 = ; 56 8 7π 4 "     −1 # ∞ X 1 9 G ζ (k) − 1 5 k 57 5γ 1 = − − + log π 2 · A 10 · 0 ; k+1 4 40 8 5π 4

(682)

=

(683)

(684)

k=2

  ∞ X ζ (k) − 1 7 k k=2

k+1

4

   9 81 7γ G 1 − 21 − 12 = − + + log 2 · π · A 14 · 0 ; 56 8 7π 4

  ∞ X ζ (2k) − 1 5 2k k=1

2k + 1

4

  ∞ X ζ (2k) − 1 7 2k k=1

2k + 1

4

(685)

=

h 1 i 4 G 3 − + log 2− 4 · 3− 5 ; 2 5π

(686)

=

h 1 2 i 2 3 G + + log 2− 4 · 3 7 · 11− 7 ; 2 7π

(687)

  ∞ X ζ (2k + 1) − 1 5 2k k=1

k+1

4 "

  − 8 # 5 2 32 4 36 3 1 = − − γ + log 2 5 · 3 25 · π 5 · A 25 · 0 ; 25 4

(688)

  ∞ X ζ (2k + 1) − 1 7 2k k=1

k+1

4 "

   8 # 7 2 16 16 4 36 3 1 . = − − γ + log 2− 7 · 3− 49 · 11 49 · π − 7 · A 49 · 0 49 4

(689)

342

Zeta and q-Zeta Functions and Associated Series and Integrals 1 4

Setting z = we obtain

in (512) through (515) and using 1.4(6), 1.1(12), 1.4(23) and 1.4(24),

∞ X (2k − 1)ζ (k) k=2

(k + 1) · 22k

=−

   1 3 7 1 γ G + + log 2− 12 · π − 2 · A− 2 · 0 ; 8 π 4

"   −1 # ∞ k X 1 1 3 γ G 1 k (2 − 1)ζ (k) (−1) = + + log 2 12 · π 2 · A 2 · 0 ; 2k 8 π 4 (k + 1) · 2

(690)

(691)

k=2

∞ X (22k − 1)ζ (2k) k=1

(2k + 1) · 24k

=

G 1 − log 2; π 4

∞ X (22k+1 − 1)ζ (2k + 1) k=1

(k + 1) · 24k

(692) "

− 83

= −γ + log 2

·π

−4

−12

·A

  8 # 1 · 0 . 4

(693)

If we set z = 14 and z = 34 in (564) through (567), by virtue of 1.4(6), 1.1(12), 1.4(23) and 1.4(24), we obtain ∞ X (−1)k k=2

h 3 1 9i γ G ζ (k) = −1 + + + log 2− 2 · π 2 · A 2 ; 8 π k(k + 1) · 22k

 k ∞ h 3 i X 1 3 3 3γ G k ζ (k) (−1) = −1 + − + log 2− 2 · 3 · π 2 · A 2 ; k(k + 1) 4 8 3π

(694)

(695)

k=2

∞ X k=2 ∞ X k=2 ∞ X k=1 ∞ X k=1 ∞ X k=1 ∞ X k=1

h 1 1 i 9 ζ (k) γ G = − + + log 2 2 · π 2 · A− 2 ; 2k 8 π k(k + 1) · 2 ζ (k) k(k + 1)

 k h 1 1 i 3 G 3 3γ − + log 2 2 · π 2 · A− 2 ; =− 4 8 3π

π  ζ (2k) 2G = −1 + + log ; π 2 k(2k + 1) · 24k ζ (2k) k(2k + 1)

 2k   3 2G 3π = −1 − + log ; 4 3π 2

i h ζ (2k + 1) 8 −36 = 4 − γ + log 2 · A ; (k + 1)(2k + 1) · 24k ζ (2k + 1) (k + 1)(2k + 1)

 2k h 8 i 4 3 4 = − γ + log 2 3 · 3− 3 · A−4 . 4 3

(696)

(697)

(698)

(699)

(700)

(701)

Series Involving Zeta Functions

343

Setting z = 41 , z = 34 , z = 54 and z = 1.1(12), 1.4(23) and 1.4(24), we get

7 4

in (573) through (576) and using 1.4(6),

∞ X (−1)k

i h 23 1 9 ζ (k) − 1 17 γ G 5 −2 2 ·A2 ; · 5 · π = − (702) + + + log 2 8 8 π k(k + 1) · 22k k=2   ∞ h 37 i X 7 1 3 ζ (k) − 1 3 k 19 3γ G (−1)k =− + − + log 2− 6 · 3 · 7 3 · π 2 · A 2 ; k(k + 1) 4 8 8 3π k=2

(703)  k h i 11 18 1 9 ζ (k) − 1 5 113 5γ G (−1)k =− + + + log 2− 2 · 3 5 · 5 · π 2 · A 10 ; k(k + 1) 4 40 8 5π

∞ X k=2

(704) ∞ X k=2

  ζ (k) − 1 7 k (−1)k k(k + 1) 4

h 11 3 i 9 11 1 G 185 7γ + − + log 2− 2 · 3 7 · 7 · 11 7 · π 2 · A 14 ; 56 8 7π ∞ h 13 i X 9 1 ζ (k) − 1 Gπ 7 γ 2 · 3−3 · π 2 · A− 2 ; − + log 2 = − 8 8 + k(k + 1) · 22k k=2   ∞ h 7 1 i X 3 ζ (k) − 1 3 k 5 3γ G =− − − + log 2 6 · π 2 · A− 2 ; k(k + 1) 4 8 8 3π k=2   ∞ h 1 1 i X ζ (k) − 1 5 k 9 7 5γ G =− − + + log 2 2 · π 2 · A− 10 ; k(k + 1) 4 40 8 5π k=2   ∞ h 1 1 i X 9 ζ (k) − 1 7 k 17 7γ G − − + log 2 2 · π 2 · A− 14 ; = k(k + 1) 4 56 8 7π

(705)

=−

(706)

(707)

(708)

(709)

k=2

∞ X k=1

h i ζ (2k) − 1 2G = −3 + + log 2−5 · 3−3 · 55 · π ; 4k π k(2k + 1) · 2

  ∞ X ζ (2k) − 1 3 2k k=1

k(2k + 1)

4

  ∞ X ζ (2k) − 1 5 2k k=1

k(2k + 1)

4

  ∞ X ζ (2k) − 1 7 2k k=1 ∞ X k=1

k(2k + 1)

4

(710)

= −3 −

h i 7 2G + log 2−5 · 3 · 7 3 · π ; 3π

(711)

= −3 +

i h 18 2G + log 2−5 · 3 5 · 5 · π ; 5π

(712)

= −3 −

h i 3 11 2G + log 2−5 · 3 7 · 7 · 11 7 · π ; 7π

(713)

h i ζ (2k + 1) − 1 72 −12 −20 −36 = 5 − γ + log 2 · 3 · 5 · A ; (k + 1)(2k + 1) · 24k

(714)

344

Zeta and q-Zeta Functions and Associated Series and Integrals

  ∞ X ζ (2k + 1) − 1 3 2k = (k + 1)(2k + 1) 4 k=1   ∞ X ζ (2k + 1) − 1 5 2k = (k + 1)(2k + 1) 4 k=1   ∞ X ζ (2k + 1) − 1 7 2k = (k + 1)(2k + 1) 4 k=1

i h 88 4 28 7 − γ + log 2 9 · 3− 3 · 7− 9 · A−4 ; 3

(715)

i h 24 72 4 36 53 − γ + log 2 5 · 3− 25 · 5− 5 · A− 25 ; 25

(716)

h 24 i 12 4 44 36 101 − γ + log 2 7 · 3− 49 · 7− 7 · 11− 49 · A− 49 . 49 (717)

Setting z = we obtain

1 4

in (590) through (593) and using 1.4(6), 1.1(12), 1.4(23) and 1.4(24),

∞ X (2k − 1)ζ (k)

k(k + 1) · 22k

k=2

=−

h 1 i γ G − + log 2 12 · A3 2 ; 8 π

(718)

∞ h 11 i X 3 (2k − 1)ζ (k) γ G 12 · A− 2 ; (−1)k = − + log 2 8 π k(k + 1) · 22k

(719)

k=2

∞ X (22k − 1)ζ (2k)

k(2k + 1) · 24k

k=1

=−

2G + log 2; π

∞ X (22k+1 − 1)ζ (2k + 1)

(k + 1)(2k + 1) · 24k

k=1

(720)

h 10 i = −γ + log 2− 3 · A12 .

(721)

Further Evaluation by Using the Triple Gamma Function We, first, rewrite the left-hand side of (488) as follows: ∞ X

ζ (2n − 1, a)z2n n = [ψ(a) − 1]z2 + (a − 1) log[0(a + z)0(a − z)]

(722)

n=2

− log[G(a + z)G(a − z)] + 2(1 − a) log 0(a) + 2 log G(a)

(|z| < |a|).

By differentiating both sides of (4) with respect to t and multiplying the resulting equation by t3 , if we integrate the resulting equation with respect to t from t = 0 to t = z, we obtain ∞ X n=3

z2n ζ (2n − 3, a) =− n

Zz 0

Z−z ψ(a) 4 t ψ(a + t)dt − t3 ψ(a + t)dt + z 2 3

(|z| < |a|),

0

(723)

Series Involving Zeta Functions

345

which, by virtue of (481), yields ∞ X

ζ (2n − 3, a)

n=3

z2n n



   3 2 7 9 3 = a − a+ + − a log(2π) − 2 log A z2 2 2 4 2 1 + [2ψ(a) − 3]z4 + (a − 1)3 log[0(a + z)0(a − z)] 4 − (3a2 − 9a + 7) log[G(a + z)G(a − z)] − 2(z − 2a + 3) log 03 (a + z) + 2(z + 2a − 3) log 03 (a − z)

(724)

+ 2(1 − a)3 log 0(a) + 2(3a2 − 9a + 7) log G(a) Zz + 4(3 − 2a) log 03 (a) + 2 log 03 (t + a)dt 0

Z−z

log 03 (t + a)dt

+2

(|z| < |a|).

0

Setting a = 2 in (722) and applying 1.3(50) and 2.3(9), we obtain ∞ X 1 z2n = −γ z2 + [1 − ζ (3)]z4 {ζ (2n − 1) − 1} n 2 n=3

(725)

+ log[0(2 + z)0(2 − z)] − log[G(2 + z)G(2 − z)] (|z| < 2). Setting z =

3 2

in (725) and making use of 1.4(8) and 2.3(9), we obtain

  ∞ X 1 3 2n n=3

n

2

{ζ (2n − 1) − 1} =

 1  73 9γ 81 − + 3 log A − ζ (3) + log 2− 12 · 5 . 32 4 32 (726)

Setting a = 4 in (722) and applying 1.3(50), 2.3(9) and 2.3(12), we obtain ∞  X ζ (2n − 1) − 1 −

    z2n 5 1 251 2 = − γ z + − ζ (3) z4 6 2 216 22n−1 32n−1 n n=3   + 3 log[0(4 + z)0(4 − z)] − log[G(4 + z)G(4 − z)] + log 2−4 · 3−6 (|z| < 4), 1

−

1



(727)

346

Zeta and q-Zeta Functions and Associated Series and Integrals

which, for z = 3, yields  ∞ 2n    X 1 1 873 81 3 ζ (2n − 1) − 1 − 2n−1 − 2n−1 = − 9γ − ζ (3) + log 3−3 · 52 . n 16 2 2 3 n=3

(728) Setting a = 4 in (724) and applying 1.3(50) and 2.3(9), we obtain ∞  X ζ (2n − 3) − 1 − n=3

1 22n−3

−



1 32n−3

z2n = n



 49 5 − log(2π) − 2 log A z2 4 2



 1 − γ z4 + 27 log[0(4 + z)0(4 − z)] − 19 log[G(4 + z)G(4 − z)] 3   (729) + 2(5 − z) log 03 (4 + z) + 2(5 + z) log 03 (4 − z) + log 2−16 3−54 + 12

Zz +2

Z−z log 03 (t + 4)dt + 2 log 03 (t + 4)dt

0

(|z| < 4).

0

Using 1.3(39), 1.4(73) and (472), we find that Z3

log 03 (t + 4) dt = 10

0

Z1

Z1 log(t + 1) dt + 4

0

Z1 + 19

Z1 log(t + 2) dt +

0

log 0(t + 1) dt + 12

0

log(t + 3) dt 0

Z1

Z1 log G(t + 1) dt + 3

0

log 03 (t + 1) dt

(730)

0

99 9 259 log 2 + 9 log 3 + log π − 24 log A + log B = −33 + 8 8 2 and Z−3 log 03 (t + 4) dt 0

Z1 =− 0

log 0(t + 1) dt − 3

Z1

Z1 log G(t + 1) dt − 3

0

3 9 9 = − log(2π) + 6 log A − log B. 4 8 2

0

log 03 (t + 1) dt

(731)

Series Involving Zeta Functions

347

If we set z = 3 in (729) and make use of (730) and (731), we obtain  ∞ 2n  X 1 1 3 ζ (2n − 3) − 1 − 2n−3 − 2n−3 n 2 3 n=3   237 81 = − γ − 54 log A + log 212 · 3−27 · 58 . 4 2

(732)

Setting a = 2 in (724) and applying 1.3(50) and 2.3(9), we obtain   ∞ X z2n 5 1 [ζ (2n − 3) − 1] = − log(2π) − 2 log A z2 n 4 2 n=3

1 − (1 + 2γ )z4 + log [0(2 + z)0(2 − z)] − log [G(2 + z)G(2 − z)] 4 + 2(1 − z) log 03 (2 + z) + 2(1 + z) log 03 (2 − z) Zz Z−z + 2 log 03 (t + 2)dt + 2 log 03 (t + 2)dt (|z| < 2). 0

(733)

0

Making use of 1.3(39), 1.4(55), 1.4(73), 1.4(77), (472) and (473), we find that 3

Z2

log 03 (t + 2) dt =

0

Z1

Z1 log G(t + 1) dt +

0

0

12 log

Z

12 log

0(t + 1) dt + 2

+

log 03 (t + 1) dt

0

12

Zlog

Z G(t + 1) dt + 0

(734) 03 (t + 1) dt

0

29 9 15 5 15 259 =− − log 2 + log π − log A − log B + log C 768 1920 16 16 4 16 and 3

Z− 2

log 03 (t + 2) dt = −2

0

Z1

12

log 03 (t + 1) dt + 12

Z1 log G(t + 1) dt − 0

0

G(t + 1) dt − 0

12

+

Z1

Zlog

Zlog

03 (t + 1) dt

0

0

+

Zlog

log 0(t + 1) dt

0

1

0(t + 1) dt −

Z− 2 log(t + 1) dt 0

29 29 3 1 15 = − log 2 − log A − log B + log C. 768 1920 16 2 16

(735)

348

Zeta and q-Zeta Functions and Associated Series and Integrals

Setting z =

3 2

in (733) and applying 1.4(8), 1.4(96), (734) and (735), we obtain

  ∞ X ζ (2n − 3) − 1 3 2n n=3

n

2

(736)

 269  1796 81 27 15 γ+ log A + log C + log 2− 480 · 5 . =− − 32 4 4

Applications of Corollary 3.3 Upon setting n = 1, 2, 3 and 4 in 3.2(62), if we make use of the appropriate identities given in previous sections, we obtain the following known or new formulas for closedform evaluations of several families of series involving the Hurwitz (or generalized) Zeta function ζ (s, a): ∞ X ζ (k, a) k+1 t = ζ 0 (−1, a) − ζ 0 (−1, a − t) k(k + 1) k=2 (737)   1 1 t2 + − a − log 0(a) + log(2π) t + [1 + ψ(a)] (|t| < |a|); 2 2 2 ∞ X ζ (k, a) tk+2 k(k + 1)(k + 2) k=2      1 2 1 1 0 0 0 a −a+ + ζ (−1, a) t (738) = ζ (−2, a − t) − ζ (−2, a) + 2 4 6   2 3 t t3 + − 3a + log(2π) − 2 log 0(a) + [3 + 2ψ(a)] (|t| < |a|); 2 2 12 ∞ X ζ (k, a) 1 tk+3 = [ζ 0 (−3, a) − ζ 0 (−3, a − t)] k(k + 1)(k + 2)(k + 3) 6 k=2     1 3 3 2 1 t 0 − a − a + a + 3 ζ (−2, a) 3 2 2 6     2 5 2 1 t (739) + a −a+ + 6 ζ 0 (−1, a) 2 6 12   3 t 11 + − 11 a + 3 log(2π) − 6 log 0(a) 2 36

+ [11 + 6 ψ(a)] ∞ X

t4 144

(|t| < |a|);

ζ (k, a) 1 tk+4 = [ζ 0 (−4, a − t) − ζ 0 (−4, a)] k(k + 1)(k + 2)(k + 3)(k + 4) 24 k=2     1 4 1 t + a − 2a3 + a2 − + 4 ζ 0 (−3, a) 4 30 24

Series Involving Zeta Functions

349

    2 7 3 3 2 1 t 0 − a − a + a + 12 ζ (−2, a) 3 2 2 48     3 1 t + 13 a2 − a + + 24 ζ 0 (−1, a) 6 144 + [25 − 50 a + 12 log(2π) − 24 log 0(a)] + [25 + 12 ψ(a)]

t5 1440

t4 576

(|t| < |a|).

(740)

Setting a = 1 in (740) and applying the formulas presented in this and previous sections, we readily have ∞ X

  3 ζ (5) ζ (k) 1 ζ 0 (−4, 1 − t) − tk+4 = k(k + 1)(k + 2)(k + 3)(k + 4) 24 4π 4 k=2    3  t ζ (3) 2 25 t 5 + 4 log C + t + − 24 log A − 72 24 16 π 2 6 144 + [12 log(2π) − 25]

t4 t5 + (25 − 12 γ ) 576 1440

(741)

(|t| < 1),

which, for t = 12 , yields ∞ X k=2

ζ (k) k(k + 1)(k + 2)(k + 3)(k + 4) · 2k

1 γ 1 4 ζ (3) 31 ζ (5) = log(2π) − − log A − log C + − . 48 240 3 3 4π2 32 π 4

(742)

Similarly, we obtain ∞ X k=2

ζ (k) k(k + 1)(k + 2)(k + 3) · 2k

1 γ 5 ζ (3) 59 ; log 2 + log π − − log A − log C + = 720 12 48 2 2π2 ∞ X ζ (k) k(k + 1)(k + 2) · 2k k=2

3 1 γ 7 ζ (3) = − + log(2π) − − 2 log A + . 8 2 12 8π2

(743)

(744)

We remark in passing that the various series identities presented here are potentially useful in deriving further identities for series involving the Zeta and related functions.

350

Zeta and q-Zeta Functions and Associated Series and Integrals

For example, if we replace t in (737) by −t, we get ∞ X ζ (k, a) k+1 (−1)k t k(k + 1) k=2

= ζ 0 (−1, a − t) − ζ 0 (−1, a) + − [1 + ψ(a)]

t2 2



 1 1 − a − log 0(a) + log(2π) t 2 2

(745)

(|t| < |a|).

Furthermore, by making use of 2.2(17) and 2.2(33), the series identity (745) can be written in an equivalent form involving the double Gamma function 02 (cf. Equation (561)): ∞ X ζ (k, a) k+1 (−1)k t k(k + 1) k=2

= (1 − a) log 0(a) − log 02 (a) + (t + a − 1) log 0(a + t) + log 02 (a + t) (746)   1 t2 1 − a + log(2π) − log 0(a) t − [ψ(a) + 1] (|t| < |a|), + 2 2 2 which, in the form (561), was proven (in a markedly different way) by Choi and Srivastava [288, p. 108, Eq. (2.14)].

3.5 Use of Hypergeometric Identities Al-Saqabi et al. [21] presented a systematic account of several interesting infinite series expressed in terms of the Psi (or Digamma) function. Aular de Dura´ n et al. [79] examined rather systematically the sums of numerous interesting families of infinite series with or without the use of fractional calculus (see also [1237]). Shen [1024] investigated the connections between the Stirling numbers s(n, k) of the first kind and the Riemann Zeta function ζ (n) by means of the Gauss summation formula 1.5(7) for the hypergeometric series. We show how the sums of certain interesting classes of infinite series can be evaluated by analyzing three known identities involving hypergeometric series. Our presentation in this section is based essentially on the works of Shen [1024], Choi and Seo [282] Choi and Srivastava [290] and Choi et al. [309]. In addition to Gauss’s summation formula 1.5(7), we, first, recall some well-known summation formulas for hypergeometric series. Gauss’s formula:      0 1 0 1 + 1a + 1b  2 2 2 2 1 1    =  2 F1 a, b; (a + b + 1); 1 1 1 2 2 (1) 0 + a 0 + 1b 2

2

2

2

(a + b 6= −1, −3, −5, . . .).

Series Involving Zeta Functions

351

Kummer’s formulas:     21−c 0(c)0 12 1    =  2 F1 a, 1 − a; c; 2 0 12 c + 21 a 0 12 c − 21 a + 21   0(1 + a − b)0 1 + 12 a  2 F1 (a, b; 1 + a − b; −1) =  0 1 + 12 a − b 0(1 + a)


c 6∈ Z− 0 ,

(2)

(3)


<(b) < 1; 1 + a − b 6∈ Z− 0 . Dixon’s theorem: " # a, b, c; 1 3 F2 1 + a − b, 1 + a − c;     0 1 + 12 a 0(1 + a − b)0(1 + a − c)0 1 + 12 a − b − c     = 0(1 + a)0 1 + 12 a − b 0 1 + 12 a − c 0(1 + a − b − c)     1 < a − b − c > −1 . 2 Watson’s theorem:   a, b, c; 1 3 F2  1 1 + a + 12b, 2c; 2  2   

   0 c + 12 0 12 + 12 a + 12 b 0 12 − 21 a − 12 b + c       . =  0 12 + 12 a 0 12 + 12 b 0 12 − 12 a + c 0 12 − 12 b + c 0

1 2

(4)

(5)

Whipple’s theorem: 3 F2 [a, b, c; d, e; 1]

0(d)0(e)        = π 21−2c  1 1 1 0 2 a + 2 e 0 2 a + 12 d 0 12 b + 12 e 0 12 b + 12 d

(6)

(a + b = 1; d + e = 1 + 2c).

Series Derivable from Gauss’s Summation Formula 1.4(7) Consider a as a variable in 1.5(7) and write f (z) := 1 +

∞ X (b)n (z)n 0(c) 0(c − b − z) = . (c)n n! 0(c − b) 0(c − z) n=1

(7)

352

Zeta and q-Zeta Functions and Associated Series and Integrals

By making use of the definition and properties of the Stirling numbers s(n, k) of the first kind, we find that ( (2) ∞ 2 X − Hn−1 3 (b)n z Hn−1 2 Hn−1 + z + z f (z) = 1 + (c)n n n 2n n=1 (8) ) (2) (3) 3 Hn−1 − 3 Hn−1 Hn−1 + 2 Hn−1 4 + z + ··· . 6n Conversely, we let ∞

X 0(c) 0(c − b − z) := 1 + f (z) = an zn , 0(c − b) 0(c − z)

(9)

n=1

where the coefficients an are to be determined. Differentiating (9) logarithmically, we obtain ∞

X f 0 (z) := cn zn , f (z) n=0

which, in view of 1.3(1) and 1.3(53), yields ( ψ(c) − ψ(c − b) (n = 0), cn = ζ (n + 1, c − b) − ζ (n + 1, c) (n ∈ N),

(10)

where ψ(z) and ζ (s, a) denote the Digamma (or Psi) and Hurwitz (or generalized) Zeta functions, respectively (see Sections 1.3 and 2.2). Observe that a0 = f (0) = 1

and c0 = f 0 (0)f (0) = a1 .

Since ∞ X n=0

! ∞ n 0 (z) X X f = an−k ck zn , (n + 1)an+1 zn = f 0 (z) = f (z) · f (z) n=0

k=0

we have the following recursion formula for an : (n + 1)an+1 =

n X

an−k ck

(n ∈ N).

(11)

k=0

Now, by comparing the power series in (8) and (9), we get ∞ X (b)n = ψ(c) − ψ(c − b) n (c)n n=1


<(c − b) > 0; c 6∈ Z− 0 ,

(12)

Series Involving Zeta Functions

353

which is a well-known (rather classical) result; ∞ i X 1h (b)n Hn−1 = {ψ(c) − ψ(c − b)}2 + ζ (2, c − b) − ζ (2, c) , (c)n n 2

(13)

n=1

which is the corrected version of a result in Shen [1024, p. 1397, Eq. (31)]): ∞ o 1 X 1 (b)n n 2 (2) Hn−1 − Hn−1 = [ψ(c) − ψ(c − b)]3 n (c)n 3

(14)

n=1

2 + [ψ(c) − ψ(c − b)][ζ (2, c − b) − ζ (2, c)] + [ζ (3, c − b) − ζ (3, c)]. 3 If use is made of 1.3(3), the classical result (12) is rewritten in its equivalent form:  ∞ ∞  X X (b)n 1 1 = − n (c)n n+c−b n+c n=1


<(c − b) > 0; c 6∈ Z− 0 .

(15)

n=0

Choose c = 1 and b = z in (15) and expand both sides as power series about z = 0. We, thus, obtain !  X ∞ ∞  ∞ ∞ ∞ X X X 1 1 (−1)n+k s(n, k) k X (z)n z = = − = ζ (k + 1) zk , n · n! n · n! n−z n k=1

n=1

n=k

n=1

k=1

which, upon equating the coefficients of zk , yields an interesting relation between the Riemann Zeta function and the Stirling numbers of the first kind: ζ (k + 1) =

∞ X (−1)n+k s(n, k) n · n!

(k ∈ N).

(16)

n=k

Similarly, we can rewrite (13) in the following form: ( )2 ∞ ∞  X (b)n Hn−1 1 X 1 1 = − (c)n n 2 n+c−b n+c n=1

n=0

+

∞  X n=0

1 − 1(n + c)2 (n + c − b)2

#

(17) ,

which, upon choosing c = 1 and b = z and expanding both sides as power series about z = 0, yields ! (∞  ) ∞ X X Hn−1 2 Hn−1 z+ z2 + · · · n n2 n=1 n=1 i 1h = ζ (3) z + {ζ (2)}2 + 3 ζ (4) z2 + · · · . 2

354

Zeta and q-Zeta Functions and Associated Series and Integrals

Therefore, we have (see Shen [1024, p. 1398]) ∞ X Hn−1 = ζ (3), n2

(18)

n=1

which is an obvious special case of 2.3(54) when n = 2:  ∞  X Hn−1 2 n=1

n

i 11 1h {ζ (2)}2 + 3 ζ (4) = ζ (4). 2 4

=

(19)

If we set (b = 1 and c = 2) and (b = 12 and c = 1) in (13) and (14) and use 1.3(50), 1.3(51), 2.3(2) and 2.3(9), we also obtain the following interesting formulas: ∞ X Hn−1 = 1; n(n + 1)

n=1 ∞ X n=1

∞ X n=1 ∞ X n=1

(20)

(2)

2 Hn−1 − Hn−1

n(n + 1)   1 2 n

n · n!  

= 2;

(21)

Hn−1 = ζ (2) + 2 (log 2)2 ;

1 2 n

o n 8 (2) 2 Hn−1 − Hn−1 = 4 ζ (3) + 4 ζ (2) log 2 + (log 2)3 . n · n! 3

(22)

(23)

Series Derivable from Kummer’s Formula (3) Consider a as a variable in (3) and apply the same technique as above. We readily obtain ∞ X (−1)n (b)n n=1

n(1 − b)n

=

1 [γ + ψ(1 − b)] , 2

(24)

which is, in fact, an immediate consequence of a known result recorded by Aular de Dura´ n et al. [79, p. 746, Eq. (2.9)] with z = 1 − b; ∞ X (−1)n (b)n n=1

n(1 − b)n

Hn−1 +

n X k=1

1 b−k

!

1 3 = [γ + ψ(1 − b)]2 + [ζ (2, 1 − b) − ζ (2)]; 8 8

(25)

Series Involving Zeta Functions ∞ X (−1)n (b)n



n(1 − b)n



n=1

355

n X k=1

1 + Hn−1 b−k

!2 +

n X k=1

 1 (2) − Hn−1  (b − k)2

1 3 = [γ + ψ(1 − b)]3 + [γ + ψ(1 − b)][ζ (2, 1 − b) − ζ (2)] 24 8 7 + [ζ (3) − ζ (3, 1 − b)]. 12

(26)

By setting b = 12 in (25) and applying some elementary identities given in Sections 1.3 and 2.3, we obtain ( ) ∞ n−1 X X 3 1 (−1)n 2 2 (log 2) + ζ (2) = Hn−1 − 2 4 n 2k + 1 n=1 k=0 ( ) ∞ n−1 X X (−1)n 1 = −2 + n k(2k + 1) n=1

k=1

from which, by using the Maclaurin series of log(1 + x), we obtain n−1 ∞ X (−1)n X n=1

n

k=1

Setting b =

2 3

1 3 1 = (log 2)2 − 2 log 2 + ζ (2). k(2k + 1) 2 4

(27)

in (25) yields

)     ( ∞ n−1 X X 1 (−1)n 2 1 · −3 + n 3 n 3 n k(3k + 1) n=1 k=1 !2 √ 1 3 3 = π + 3 log 3 + [ζ (2, 13 ) − ζ (2)]. 32 3 8 Finally, by setting b = ∞ X (−1)n+1 n=1

n



1 2

(28)

in (26), we obtain the following series evaluation: n−1

X 1 2  + 2n − 1 k(2k − 1) k=1

!2 +

n X k=1

 4 (2) − Hn−1  (2k − 1)2

(29)

1 3 7 = (log 2)3 + ζ (2) log 2 + ζ (3). 3 2 2 Next, by considering b as a variable in (3), we also show that the sums of a different class of infinite series can be evaluated. For b = z, we find from (3) that   ∞ 0 1 + 21 a X (−1)n (a)n (z)n 0(1 + a − z) . f (z) := 1 + · = ·  (30) n! (1 + a − z)n 0(1 + a) 0 1 + 1 a − z n=1

2

356

Zeta and q-Zeta Functions and Associated Series and Integrals

Now, since n X k X

Bk,` =

n X n X

Bk,` ,

`=0 k=`

k=0 `=0

we can apply 1.5(5) to obtain (1 + a − z)n =

n X

(−1)n+k s(n, k)(1 + a − z)k

k=0 n X

( k   ) X k k−` ` ` = (−1) s(n, k) (1 + a) · (−1) · z ` `=0 k=0 " n #   n X X k n+k ` k−` = (−1) s(n, k) (−1) · (1 + a) z` . ` `=0

n+k

(31)

k=`

Expanding the first member of (30) in terms of the Stirling numbers s(n, k) and using 1.6(9), 1.6(11) and (31), we have f (z) = 1 +

∞ X

  (−1)n (a)n rn! α1 z + α2 z2 + α3 z3 + · · · ,

(32)

n=1

where (n − 1)! ; (1 + a)n (n − 1)! α2 = (Hn−1 + ψ(1 + a + n) − ψ(1 + a)); (1 + a)n  (n − 1)! (2) α3 = −Hn−1 + (Hn−1 + ψ(1 + a + n) − ψ(1 + a))2 2(1 + a)n  + ζ (2, 1 + a) − ζ (2, 1 + a + n)

α1 =

and so on. Now, using the same technique as above, we obtain   ∞ X (−1)n a 1 = ψ 1 + a − ψ(1 + a), n(n + a) 2

(33)

n=1

which was recorded by Al-Saqabi et al. [21, p. 366, Eq. (2.7)], who commented that (33) is essentially the same as the known result (cf. Hansen [531, p. 102, Eq. (6.1.14)]): ∞ X n=1

    (−1)n 1 λ = β 1+ − log 2 , n(λ + µn) λ µ

(34)

Series Involving Zeta Functions

357

where β(z) is defined by β(z) :=

     X ∞ 1 1 1 1 (−1)N ψ + z −ψ z = ; 2 2 2 2 z+N

(35)

N=0

( ) n ∞ X X 1 (−1)n a Hn−1 + n(n + a) k+a n=1

k=1

(36)

"   2  # 1 1 1 ψ 1 + a − ψ(1 + a) + ζ (2, 1 + a) − ζ 2, 1 + a ; = 2 2 2   !2 n n ∞  X X 1 1 (−1)n a X (2) − Hn−1 + + Hn−1 2  n(n + 1)  k+a (k + a) n=1

k=1

k=1

    h   2 1 1 1 = ψ 1 + a − ψ(1 + a) ψ 1 + a − ψ(1 + a) (37) 3 2 2   i    1 2 1 + 3 ζ (2, 1 + a) − ζ 2, 1 + a + ζ (3, 1 + a) − ζ 3, 1 + a 2 3 2
− a 6∈ Z0 \ {0} . Setting a = 1 in (36) and (37) and simplifying the resulting equations with the aid of various known (or easily derivable) identities, including the following (cf., e.g., Hansen [531, p. 116, Entry (6.2.14)] for m − 2 = n = 1 and m = n − 2 = 1): ∞ X 1 3 (−1)n+1 = 2 log 2 − 3 + ζ (2) + ζ (3); 2 4 n(n + 1)3 n=1

∞ X (−1)n+1 1 3 = 2 log 2 − 1 − ζ (2) + ζ (3); 2 4 n3 (n + 1)

(38)

n=1

∞ ∞ X X (−1)n+1 (−1)n 3 H = 4 Hn + ζ (2) + ζ (3), n 2 2 n 4 n (n + 1) n=1

n=1

we readily obtain ∞ X (−1)n 1 Hn = (log 2)2 − ζ (2), n(n + 1) 2

(39)

n=1

which reduces immediately to ∞ i X (−1)n+1 1h Hn = ζ (2) − (log 2)2 . n 2 n=1

(40)

358

Zeta and q-Zeta Functions and Associated Series and Integrals

We can also readily obtain the following identity: ∞ X 7 2 1 (−1)n 2 Hn = − − (log 2)3 + ζ (2) log 2 − ζ (3). n(n + 1) 9 3 3

(41)

n=1

Series Derivable from Other Hypergeometric Summation Formulas Consider a as a variable in (2) and apply the same technique as above. We, thus, find that ∞ X (n − 1)!

     1 1 1 = ψ 12c + −ψ c ; (42) n (c)n · 2 2 2 2 n=1 "         2 # ∞ X (n − 1)! 1 1 1 1 1 1 1 = ζ 2, c + ζ 2, c + − ψ c+ −ψ c , n (c)n · 2n 8 2 2 2 2 2 2 n=1

(43) which, upon setting c = 1, 2 and 3, yields ∞ X n=1

i 1h 1 2 = ζ (2) − (log 2) , 2 n2 · 2n

(44)

which may be compared with (40): ∞ X n=1 ∞ X n=1

1 = ζ (2) − 1 − (log 2 − 1)2 ; n2 (n + 1) · 2n−1

(45)

1 58 1 1n2 (n + 1)(n + 2) · 2n−1 = ζ (2) − (2 log 2 − 1)2 . 2 −8

(46)

By setting c = z in (2) and 1.5(7), multiplying the resulting equations by z and making use of the same technique as above, we obtain the following summation formula:     ∞ X 0(a + n) 0(1 − a + n) 1 1 1 1 =√ 0 + a 0 1− a , n! (n − 1)! 2n 2 2 2 π

(47)

n=1

which, after some elementary manipulation, becomes     1 1 1   1 2 0 2 + 2a 0 1 − 2a =√ . 2 F1 1 + a, 2 − a; 2; 2 0(1 + a) 0(2 − a) π

(48)

Series Involving Zeta Functions

359

By using Legendre’s duplication formula 1.1(29), (48) is rewritten in its equivalent form: √   π 1   ,  F 1 + a, 2 − a; 2; = 2 1 3 2 − 12 a 0 1 + 12 a 0 r2

(49)

which can be shown to be a special case of Kummer’s summation formula (2) in view of Euler’s formula 1.5(21): ∞ X 0(a + n) 0(1 − a + n)

n! (n − 1)! 2n

n=1

Hn−1 = 0(a) 0(1 − a)

(50)         1 1 1 1 1 1 1 1 1 + a 0 1 − a log 2 + γ + ψ a + ψ − a ; +√ 0 2 2 2 2 2 2 2 2 π

∞ o X 0(a + n) 0(1 − a + n) n 2 (2) H + H n−1 n−1 n! (n − 1)! 2n n=1         2 1 1 1 1 1 1 1 1 1 =√ 0 + a 0 1− a log 2 + γ + ψ a + ψ − a 2 2 2 2 2 2 2 2 π      1 1 1 1 + ζ (2) − ζ 2, a + ζ 2, − a ; (51) 4 2 2 2 ∞ X (a)n (b)n 0(−a − b) = (n − 1)! n! 0(−a) 0(−b)

(<(a + b) < 0),

(52)

n=1

which, after some elementary manipulation, is seen to be the same as the following special case of Gauss’s summation theorem 1.5(7): 2 F1 (a + 1, b + 1; 2; 1) =

0(−a − b) 0(1 − a)0(1 − b)

(<(a + b) < 0);

(53)

∞ X 0(−a − b) (a)n (b)n [γ + ψ(−a) + ψ(−b) − ψ(−a − b)] ; Hn−1 = 1 + (n − 1)! n! 0(−a) 0(−b) n=1

(54) ∞ o X (a)n (b)n n 2 0(−a − b) (2) Hn−1 + Hn−1 = (n − 1)! n! 0(−a)0(−b) n=1

h · {γ + ψ(−a) + ψ(−b) − ψ(−a − b)}2 + ζ (2) i + ζ (2, −a − b) − ζ (2, −a) − ζ (2, −b) .

(55)

360

Zeta and q-Zeta Functions and Associated Series and Integrals

Setting a = 12 in (47), (50) and (51) and simplifying with the aid of various known identities involving Gamma and Psi functions, we obtain ∞ X n=1

{(2n)!}2 1 n  3 o2 ; = 0 4 √ (n!)3 (n − 1)! 25n π π

∞ X

{(2n)!}2

n=1

(n!)3 (n − 1)! 25n

∞ X n=1

(56)

 1 n  o2  π Hn−1 = 1 − √ 0 34 + 2 log 2 ; 2 π π

(57)

n o {(2n)!}2 (2) 2 H + H n−1 n−1 (n!)3 (n − 1)! 25n

=

 2  1 n  o2  π 1  + 2 log 2 + ζ (2) − ζ 2, 41 . √ 0 34 2 2 π π

(58)

In view of Euler’s formula 1.5(21), the summation formula (56) can be deduced directly from (2), by setting a = 12 and c = 2. Moreover, (56) follows readily from the special case of the identity (1) when a = b = 23 . Considering (1) just as above, we obtain ∞ X

    (b)n 1 1 1   =ψ + b −ψ , 1 1 2 2 2 n−1 + b n=1 n · 2 2 2

(59)

n

which is precisely the same as the aforementioned known result recorded (with correction) by Al-Saqabi et al. [21, p. 362, Eq. (1.7)]: ∞ X

(b)n   n−1 1 + 1 b n=1 n · 2 2 2

2 Hn−1 − n

n−1 X

1

k=0

k + 12 + 12 b

! (60)

  2        1 1 1 1 1 1 1 1 + b −ψ = ψ + ζ 2, + b − ζ 2, . 4 2 2 2 4 2 2 2 If we consider a as a variable in Dixon’s theorem (4), similarly as above, we obtain ∞ X n=1

1 (b)n (c)n = {γ + ψ(1 − b) + ψ(1 − c) − ψ(1 − b − c)} , n(1 − b)n (1 − c)n 2

(61)

Series Involving Zeta Functions

361

which is an immediate consequence of the special case a = 0 of a known result recorded by Srivastava [1079, p. 82, Eq. (3.2)]: " n  #  ∞ X 1 X 1 1 1 (b)n (c)n − − − n (1 − b)n (1 − c)n k k−b k−c n n=1 k=1 (62) 1 = [γ + ψ(1 − b) + ψ(1 − c) − ψ(1 − b − c)]2 8 + 38 [ζ (2, 1 − b) + ζ (2, 1 − c) − ζ (2) − ζ (2, 1 − b − c)] ; ( )2  ∞ n  X X 1 1 1 (b)n (c)n 1  + − + n (1 − b)n (1 − c)n k−b k−c k n n=1 k=1   n  X 1 1 1 1 + + − + 2 (k − b)2 (k − c)2 k2 n k=1 (63) 1 [γ + ψ(1 − b) + ψ(1 − c) − ψ(1 − b − c)] = 24 · [{γ + ψ(1 − b) + ψ(1 − c) − ψ(1 − b − c)}2 + 3 {ζ (2, 1 − b) + ζ (2, 1 − c) − ζ (2) − ζ (2, 1 − b − c)}] 7 [ζ (3) − ζ (3, 1 − b) − ζ (3, 1 − c) + ζ (3, 1 − b − c)] . + 12

Further Summation Formulas Related to Generalized Harmonic Numbers As shown in preceding subsections, by employing the univariate series expansion of classical hypergeometric formulas, Shen [1024] and Choi and Srivastava [290, 298] investigated the evaluation of infinite series related to generalized harmonic numbers. More summation formulas have been systematically derived by Chu [315] and Chu and de Donno [316], who developed fully this approach to the multivariate case. For example, Chu [315] investigated the hypergeometric summation theorem due to Gauss: " # x, y 0(1 − z)0(1 − x − y − z) , (64) 1 = 2 F1 0(1 − x − z)0(1 − y − z) 1−z as well as Kummer’s and Dixon’s summation theorems. The hypergeometric series of (64) may be expanded in terms of the partial sums of the Riemann Zeta function through the symmetric functions generated by the following finite products (cf. [793]): n   x3   Y x x2  2 1+ = 1 + x Hn + Hn − Hn(2) + Hn3 − 3 Hn Hn(2) + 2 Hn(3) + · · · , k 2 6

k=1

(65)

362

Zeta and q-Zeta Functions and Associated Series and Integrals n   Y x −1 x2  2 1− = 1 + x Hn + Hn + Hn(2) k 2

k=1

(66)

 x3  3 + Hn + 3 Hn Hn(2) + 2 Hn(3) + · · · , 6   n  Y y y2  2 1+ = 1 + y On + On − O(2) n 2k − 1 2

k=1

+ n  Y 1− k=1

y 2k − 1

y3



6

(67)



(3) O3n − 3 On O(2) + ··· , n + 2 On

−1 = 1 + y On +

 y2  2 On + O(2) n 2 (68)

+

y3



6



(3) O3n + 3 On O(2) + ··· , n + 2 On

where On ’s are defined by On =

n X k=1

1 2k − 1

O(r) n

and

=

n X k=1

1 (2k − 1)r

(r ∈ N \ {1}).

(69)

Conversely, the right-hand side of (64) may be expanded as multivariate formal power series, by means of the following expansions: ∞ X σk k 0(1 − z) = exp z k

! [σ1 = γ ; σk = ζ (k) (k ∈ N \ {1})]

(70)

k=1

and !  ∞ X √ 1 τk k 0 − z = π exp z 2 k 

k=1

[τ1 = γ + 2 log 2;

(71)

τk = (2k − 1) ζ (k) (k ∈ N \ {1})].

Then, term-by-term comparison of the coefficients from two power series may yield an infinite number of summation formulas. Here, we choose to record some summation formulas involving harmonic numbers and other related numbers presented very recently by Zheng [1264], who, motivated by the work of Chu [315], established further numerous summation formulas expressing infinite series related to generalized harmonic numbers in terms of the Riemann

Series Involving Zeta Functions

363

Zeta function ζ (m) with m = 5, 6, 7. (2) (3) ∞ 3 X Hn−1 − 3 Hn−1 Hn−1 + 2 Hn−1

n2

n=1 ∞ X

(2)

8 Hn2 + 2 Hn−1 n3

n=1 ∞ X n=1

Hn−1 + Hn − n4

(2)

3 Hn Hn + 4 3 n n

! =

= 6 ζ (5). ! = 12 ζ (5).

9 ζ (5). 2

n o (2) (2) ∞ (Hn−1 + Hn )2 − Hn−1 + Hn On X n2

n=1

(72)

(73)

(74)

= 31 ζ (5).

(75)

∞ X 7π2 93 (Hn−1 + Hn )2 On = ζ (3) + ζ (5). 2 12 4 n

(76)

∞ X 93 (Hn−1 + Hn ) O2n = ζ (5). 8 n2

(77)

∞ (3) X 93 2 O3n + On ζ (5). = 2 8 n

(78)

∞ X On 7π2 31 = − ζ (3) + ζ (5). 4 12 4 n

(79)

n=1

n=1

n=1

n=1


∞ 2 X Hn−1 + Hn2 On n=1

n2

=

31 ζ (5). 2

(80)

∞ X 31 Hn−1 Hn On 7π2 ζ (3) + ζ (5). = 2 24 8 n

(81)

∞ X π4 7π2 (On−1 + On )3 2 2 = log 2 + π log 2 + ζ (3). 16 32 (2n − 1)2

(82)

∞ X On−1 + On π4 π2 = log 2 − ζ (3). 48 32 (2n − 1)4

(83)

n=1

n=1

n=1

(3) (3) ∞ X On−1 + On n=1

(2n − 1)2

=

π2 ζ (3). 8

(84)

364

Zeta and q-Zeta Functions and Associated Series and Integrals

n o ∞ (On−1 + On ) (On−1 + On )2 − O(2) + O(2) X n n−1 (2n − 1)2

n=1

π4 3π2 = log 2 + π 2 log3 2 + ζ (3). 12 16 n  o   (3) (2) ∞ (On−1 + On ) (On−1 + On )2 − 3 O(2) − O(2) + 2 On−1 + On X n n−1

(85)

(2n − 1)2

n=1

π4 3π2 = log 2 + π 2 log3 2 + ζ (3). 8 8 ∞ X π6 π4 7π 2 (On−1 + On )4 = + log2 2 + 2π 2 log4 2 + log 2ζ (3). 2 90 4 4 (2n − 1)

(86)

(87)

n=1

3.6 Other Methods and their Applications The Weierstrass Canonical Product Form for the Gamma Function Starting from the Weierstrass canonical product formula 1.1(2), Zhang and Williams [1250] proved the following well-known identity (cf. Hansen [531, p. 46, Entry (54.3.4)]): ∞ X k=1

1 ζ (2k + 1) t2k+1 = − [1 + π t cot(πt)] − t [ψ(t) + γ ] (|t| < 1), 2

(1)

which may be compared with 3.4(16). Now, for the generalized Euler constant γ (r, n), defined by

γ (r, n) := lim

   

m→∞  



m X `=1 `≡r(mod n)



    1 1 − log m  ` n  

(r ∈ Z; n ∈ N),

(2)

so that γ (0, 1) = γ ,

(3)

Lehmer [741, Theorem 7] showed that γ (r, n) = −

i 1 h r ψ + log n n n

(0 < r ≤ n; r, n ∈ N).

(4)

Series Involving Zeta Functions

365

A combination of (1) and (4) yields ∞ X k=1

ζ (2k + 1)

 r 2k n

= log n −

 rπ  π n cot + n γ (r, n) − γ − 2 n 2r

(5)

(0 < r ≤ n; r, n ∈ N), which was given by Zhang and Williams [1250, Corollary 4], who also considered its many special cases, included in Section 3.4, by evaluating γ (r, n). Next, we rewrite (1) as ∞ X k=1

ζ (2k + 1) t2k =

π 1 − cot(πt) − ψ(t + 1) − γ 2t 2

(|t| < 1),

(6)

which, upon integrating and differentiating, yields the well-known result (see Hansen [531, p. 356, Entry (54.5.8)]; Zhang and Williams [1250, Theorems 5 and 6]):   ∞ X ζ (2k + 1) 2k+1 1 πt t = log − log 0(1 + t) − γ t (0 < |t| < 1); (7) 2k + 1 2 sin π t k=1  2  ∞ X 1 π t 1 2k−1 − ψ 0 (t) (0 < |t| < 1); (8) 2k ζ (2k + 1) t = + 2 sin2 π t t2 k=1

∞ X (2k + 1) ζ (2k + 1) t2k k=1

  1 π2 t = − π cot(πt) − ψ(t) − γ − ψ 0 (t) t 2 sin2 πt The special cases of (8) and (9) when t =

1 2

(9) (0 < |t| < 1).

yield

∞ X 2k ζ (2k + 1) = 1; 22k

(10)

∞ X (2k + 1) ζ (2k + 1) = log 2. 22k+1

(11)

k=1

k=1

By replacing t in 3.4(18) by it, we obtain ∞ X k=1 ∞ X k=1

(−1)k−1 ζ (2k) t2k−1 =

π 1 coth(πt) − 2 2t

(−1)k−1 {ζ (2k) − 1} t2k−1 =

(0 < |t| < 1);

π 3t2 + 1
coth(πt) − 2 2t t2 + 1

(12) (0 < |t| ≤ 1). (13)

366

Zeta and q-Zeta Functions and Associated Series and Integrals

The special cases of (12) and (13) when t =

1 2

and t = 1 give us the following sums:

∞ π  1 X (−1)k−1 ζ (2k) π = coth − ; 2k 4 2 2 2

k=1 ∞ X

(−1)k−1 {ζ (2k) − 1} t2k−1 =

k=1

(14)

π coth π − 1. 2

(15)

By integrating (12) (or, alternatively, replacing t in 3.4(17) by it), we find that   ∞ X (−1)k−1 ζ (2k) 2k sinh πt t = log k πt

(0 < |t| ≤ 1),

(16)

k=1

which, for t = 1 and t = 21 , yields   ∞ X sinh π (−1)k−1 ζ (2k) = log ; k π k=1   ∞ X 2 sinh π2 (−1)k−1 ζ (2k) . = log π k · 22k

(17)

(18)

k=1

Evaluation by Using Infinite Products A remarkable infinite product formula was recently posed as a problem by Wilf [1228]. Subsequently, Choi and Seo [282] proved Wilf’s formula, as well as three other analogous product formulas. Choi et al. [275], as a sequel to these earlier works, presented several general infinite product formulas, which include, as their special cases, the aforementioned product formulas of Wilf [1228] and Choi and Seo [282]. Some other related results, series involving the Riemann Zeta function, are also considered briefly. In 1997, Wilf [1228] posed as a problem the following elegant infinite product formula, which contains some of the most important mathematical constants, such as π , e and the Euler-Mascheroni constant γ , defined by 1.1(3): ∞  Y j=1

− 1j

e

 1+

1 1 + 2 j 2j

π

 =

π

e 2 + e− 2 . π eγ

(19)

Subsequently, Choi and Seo [282] proved (19), as well as three other similar product formulas, by making use of well-known infinite product formulas for circular and hyperbolic functions and the familiar Stirling formula 1.1(32). Choi et al. [275] presented the following two general infinite product formulas, which include Wilf’s formula (19) and other similar formulas in [282] as their

Series Involving Zeta Functions

367

special cases: !) (
∞ Y 2 eα π + e−α π 1 α 2 + 14 − 1j
1+ + = e j j2 4 α 2 + 1 π eγ j=1   √ 1 α ∈ C; α 6= ± i; i = −1 2

(20)

and   ∞  Y 2 β2 + 1 eβ π − e−β π −2
e j 1+ + = 2 j j 2β β 2 + 1 π e2γ j=1

(β ∈ C \ {0}; β 6= ±i). (21)

Indeed, we begin by letting   ∞  Y p q − pj 1+ + 2 A(p, q) := e (p, q ∈ C; <(p) > 0) j j

(22)

j=1

and take logarithms on both sides of (22). Then, using the Euler-Mascheroni constant γ given in 1.1(3), we obtain   n n   X X log A(p, q) = −p γ + lim −p log n + log j2 + pj + q − 2 log j , n→∞

j=1

j=1

which, upon applying Stirling’s formula 1.1(52) to the last summation, yields log A(p, q) = −p γ − log(2π)  + lim −(2n + p + 1) log n + 2n +

n X

n→∞

   (23) log j2 + pj + q  .

j=1

Now, we recall the infinite product formula of cosh z: cosh z =

∞  Y 1+ j=1

4z2 (2j − 1)2 π 2

 =

ez + e−z . 2

(24)

If we set z = α π in (24), take logarithms on both sides of the resulting equation and use the following version of Stirling’s formula 1.1(32): n X j=1

  1 log(2j − 1) ∼ n + log 2 − n + n log n (n → ∞), 2

368

Zeta and q-Zeta Functions and Associated Series and Integrals

we find that   n n o X
log eα π + e−α π = lim  log (2j − 1)2 + 4 α 2 + 2n − 2n log(2n) . n→∞

(25)

j=1

Combining (23) and (25), we obtain
log A(p, q) = −p γ − log(2π) + log eα π + e−α π  n   X  + lim −(p + 1) log n + log 4j2 + 4pj + 4q n→∞

(26)

j=1

 n n o X − log (2j − 1)2 + 4 α 2  . j=1

In a similar manner, if we consider the infinite product formula of sinh z, instead of cosh z in (24), we also have
log A(p, q) = −p γ − log(2βπ) + log eβ π − e−β π   n n     X X + lim −p log n + log j2 + pj + q − log j2 + β 2  . n→∞

j=1

(27)

j=1

Finally, by setting     (p, q) = 1, α 2 + 14 and (p, q) = 2, β 2 + 1 in (26) and (27), respectively, we arrive at the desired formulas (20) and (21). We note that the special case of (20) when α = 12 leads to Wilf’s formula (19). Other interesting special cases of (20) and (21) include all of the aforementioned results of Choi and Seo [282]. Taking logarithms on both sides of (20) and (21) and using the Riemann Zeta function ζ (s), defined by 2.3(1), together with the Maclaurin expansion of log(1 + x), we obtain the following interesting closed-form evaluation of two families of series involving the Riemann Zeta function:   `  j−1 n   ∞ n+1 X X X n 1 1  (−1) α2 + ζ (n + `) − n+`  n ` 4  k `=0 n=2 k=1 " !#   j−1 X 1 1 1 1 α 2 + 14 2 = + α + − log 1 + + k 4 k2 k k2 k=1

(28)

Series Involving Zeta Functions

369


!   2 eαπ + e−απ 1 2
− α + ζ (2) + log 4 4 α 2 + 1 π eγ   1 2 2 j − 1 ∈ N; α < j − j − ; α ∈ R ; 4       j−1 ` n ∞ X 1  X β2 + 1  2n X n ζ (n + `) − (−1)n+1  n 2 ` kn+`  `=0

n=2

k=1

j−1  X

  2 β2 + 1 2 β2 + 1 + + = − log 1 + k k k2 k2 (29) k=1 !   eβπ − e−βπ
− β 2 + 1 ζ (2) + log 2β β 2 + 1 πe2γ   j − 2 ∈ N; β 2 < j2 − 2j − 1; β ∈ R . The summation formulas (28) and (29), which we have presented here, are obviously different from the various classes of series involving Zeta functions, which we considered elsewhere in this chapter.

Higher-Order Derivatives of the Gamma Function Here, we present explicit formulas for the evaluation of higher-order derivatives of the familiar Gamma function and consider several applications of these explicit formulas, including evaluation of some families of definite integrals (see Choi and Srivastava [293]). It follows from 2.3(19) that 5 7 {ζ (2)}2 = ζ (4), ζ (2)ζ (4) = ζ (6), 2 6 5 7 2 ζ (2)ζ (6) = ζ (8), {ζ (4)} = ζ (8), 3 6 33 11 ζ (2)ζ (8) = ζ (10), ζ (4)ζ (6) = ζ (10). 20 10

(30)

In addition to the Maclaurin series expansion 3.4(10) for log 0(1 + z), it is also known that (cf., e.g., Luke [779, Vol. I, p. 27]) 0(1 + z) =

∞ X

an zn

(|z| < 1),

(31)

n=0

where a0 = 1

and n an = −γ an−1 +

n X k=2

(−1)k an−k ζ (k).

(32)

370

Zeta and q-Zeta Functions and Associated Series and Integrals

We, first, evaluate some higher-order derivatives of the Gamma function 0(z) at z = 1 and z = 12 . Upon replacing z by z − 1 in (31), we readily obtain an =

0 (n) (1) n!

(n ∈ N0 ).

(33)

Formula (32) and (33), together, immediately yield the recursion formula: 0 (n+1) (1) = −γ 0 (n) (1) + n!

n X (−1)k+1 k=1

(n − k)!

ζ (k + 1) 0 (n−k) (1)

(n ∈ N0 ),

(34)

which can alternatively be derived directly from the definition 1.3(52) with n = 1 (and z replaced by z + 1) by applying such results as the relationship 1.3(53) with z = 1. Similarly, we obtain another recursion formula:     (n+1) 1 (n) 1 0 = −δ 0 2 2   n  X (−1)k+1  k+1 (35) (n−k) 1 + n! 2 − 1 ζ (k + 1) 0 (n − k)! 2 k=1

(n ∈ N0 ; δ := γ + 2 log 2). Now, using the formulas in earlier sections, we compute 0 (n) (1) explicitly for the first ten derivatives: 0 0 (1) = −γ = ψ(1); 0 (2) (1) = γ 2 + ζ (2); 0 (3) (1) = −γ 3 − 3γ ζ (2) − 2ζ (3), which is the corrected version of a formula given by Campbell [208, p. 25]: 0 (4) (1) = γ 4 + 6 γ 2 ζ (2) + 8γ ζ (3) +

27 ζ (4); 2

0 (5) (1) = −γ 5 − 10 γ 3 ζ (2) − 20 γ 2 ζ (3) 135 γ ζ (4) − 20 ζ (2)ζ (3) − 24 ζ (5); 2 405 2 0 (6) (1) = γ 6 + 15 γ 4 ζ (2) + 40 γ 3 ζ (3) + γ ζ (4) 2 −

+ 120 γ ζ (2)ζ (3) + 144 γ ζ (5) + 40{ζ (3)}2 + 0 (7) (1) = −γ 7 − 21 γ 5 ζ (2) − 70 γ 4 ζ (3) −

2745 ζ (6); 8

945 3 γ ζ (4) 2

− 420 γ 2 ζ (2)ζ (3) − 504 γ 2 ζ (5) − 280 γ {ζ (3)}2 − − 945 ζ (3)ζ (4) − 504 ζ (2)ζ (5) − 720 ζ (7);

19215 γ ζ (6) 8

Series Involving Zeta Functions

371

0 (8) (1) = γ 8 + 28 γ 6 ζ (2) + 112 γ 5 ζ (3) + 945γ 4 ζ (4) + 1120 γ 3 ζ (2)ζ (3) + 1344 γ 3 ζ (5) + 1120 γ 2 {ζ (3)}2 19215 2 γ ζ (6) + 7560 γ ζ (3)ζ (4) + 4032 γ ζ (2)ζ (5) + 5760 γ ζ (7) + 2 132405 + 1120 ζ (2){ζ (3)}2 + 2688 ζ (3)ζ (5) + ζ (8); 8 0 (9) (1) = −γ 9 − 36 γ 7 ζ (2) − 168 γ 6 ζ (3) − 1701 γ 5 ζ (4) − 2520 γ 4 ζ (2)ζ (3) 57645 3 − 3024γ 4 ζ (5)−3360γ 3 {ζ (3)}2 − γ ζ (6) − 34020γ 2 ζ (3)ζ (4) 2 − 18144γ 2 ζ (2)ζ (5) − 25920 γ 2 ζ (7) − 10080 γ 2 ζ (2){ζ (3)}2 1191645 − 24192 γ ζ (3)ζ (5) − γ ζ (8) − 57645 ζ (3)ζ (6) 8 − 40824 ζ (4)ζ (5) − 25920 ζ (2)ζ (7) − 2240 {ζ (3)}3 − 40320 ζ (9); 0 (10) (1) = γ 10 + 45 γ 8 ζ (2) + 240 γ 7 ζ (3) + 2835 γ 6 ζ (4) + 5040 γ 5 ζ (2)ζ (3) 288225 4 γ ζ (6) + 6048 γ 5 ζ (5) + 8400 γ 4 {ζ (3)}2 + 4 + 113400 γ 3 ζ (3)ζ (4) + 60480 γ 3 ζ (2)ζ (5) + 86400 γ 3 ζ (7) 5958185 2 + 50400 γ 2 ζ (2){ζ (3)}2 + 120952 γ 2 ζ (3)ζ (5) + γ ζ (8) 8 + 576450γ ζ (3)ζ (6) + 408240γ ζ (4)ζ (5) + 259200γ ζ (2)ζ (7) + 22400γ {ζ (3)}3 + 403200γ ζ (9) + 113400{ζ (3)}2 ζ (4) + 120960ζ (2)ζ (3)ζ (5) + 72576{ζ (5)}2 + 172800ζ (3)ζ (7) 42329385 + ζ (10). 32 The corresponding problem for 0 (n)

  1 2

(n = 1, . . . , 10) yields

  √ 1 0 = −δ π, 2   1 (2) 1 = δ 2 + 3ζ (2), √ 0 2 π   1 (3) 1 = −δ 3 − 9δζ (2) − 14ζ (3), √ 0 2 π 0

all three of which are also recorded by Campbell [208, p. 25];   1 1 315 = δ 4 + 18δ 2 ζ (2) + 56δζ (3) + ζ (4); √ 0 (4) 2 2 π

372

Zeta and q-Zeta Functions and Associated Series and Integrals

  1 1 = −δ 5 − 30δ 3 ζ (2) − 140δ 2 ζ (3) √ 0 (5) 2 π 1575 − δζ (4) − 420ζ (2)ζ (3) − 744ζ (5); 2   1 (6) 1 4725 2 = δ 6 + 45δ 4 ζ (2) + 280δ 3 ζ (3) + δ ζ (4) √ 0 2 2 π + 2520δζ (2)ζ (3) + 4464δζ (5) + 1960{ζ (3)}2 +

131355 ζ (6); 8

  1 (7) 1 11025 3 = − δ 7 − 63δ 5 ζ (2) − 490δ 4 ζ (3) − δ ζ (4) − 8820δ 2 ζ (2)ζ (3) √ 0 2 2 π 919485 − 15624δ 2 ζ (5) − 13720δ{ζ (3)}2 − δζ (6) 8 − 77175ζ (3)ζ (4) − 46872ζ (2)ζ (5) − 91440ζ (7);   1 1 = δ 8 + 84δ 6 ζ (2) + 784δ 5 ζ (3) + 11025δ 4 ζ (4) + 23520δ 3 ζ (2)ζ (3) √ 0 (8) 2 π 919485 2 + 41664δ 3 ζ (5) + 54880δ 2 {ζ (3)}2 + δ ζ (6) + 617400δζ (3)ζ (4) 2 + 374976δζ (2)ζ (5) + 731520δζ (7) + 164640ζ (2){ζ (3)}2 25859925 + 583296ζ (3)ζ (5) + ζ (8); 8   1 (9) 1 √ 0 2 π = −δ 9 − 108 δ 7 ζ (2) − 1176 δ 6 ζ (3) − 19845 δ 5 ζ (4) − 52920 δ 4 ζ (2)ζ (3) 2758455 3 − 93744δ 4 ζ (5) − 164640 δ 3 {ζ (3)}2 − δ ζ (6) − 2778300 δ 2 ζ (3)ζ (4) 2 − 1687392 δ 2 ζ (2)ζ (5) − 3291840 δ 2 ζ (7) − 1481760 δ ζ (2) {ζ (3)}2 232739325 − 5249664 δ ζ (3)ζ (5) − δ ζ (8) − 17688510 ζ (3)ζ (6) 8 − 14764680 ζ (4)ζ (5) − 768320 {ζ (3)}3 − 9875520 ζ (2)ζ (7) − 20603520 ζ (9);   1 (10) 1 √ 0 2 π = δ 10 + 135 δ 8 ζ (2) + 1680 δ 7 ζ (3) + 33075 δ 6 ζ (4) 13792275 4 δ ζ (6) 4 + 9261000 δ 3 ζ (3) ζ (4) + 5624640 δ 3 ζ (2)ζ (5) + 10972800 δ 3 ζ (7) 1163696625 2 + 7408800 δ 2 ζ (2) {ζ (3)}2 + 26248320 δ 2 ζ (3)ζ (5) + δ ζ (8) 8 + 191471175 δ ζ (3)ζ (6) + 147646800 δ ζ (4)ζ (5) + 7683200 δ {ζ (3)}3

+ 105840δ 5 ζ (2)ζ (3) + 187488δ 5 ζ (5) + 411600δ 4 {ζ (3)}2 +

Series Involving Zeta Functions

373

+ 98755200 δ ζ (2)ζ (7) + 206035200 δ ζ (9) + 64827000 {ζ (3)}2 ζ (4) + 78744960 ζ (2)ζ (3)ζ (5) + 153619200 ζ (3)ζ (7) 33466588155 + 69745536 {ζ (5)}2 + ζ (10), 32 δ being defined already with (35).   For applications of our evaluations for 0 (n) (1) and 0 (n) 12 , by replacing z by z + 1 in 1.1(1), differentiating both sides of the resulting equation n times and then setting z = 0, we obtain 0

(n)

(1) =

Z∞

e−t (log t)n dt

(n ∈ N0 ).

(36)

0

Similarly, we have 0

(n)

  Z∞ −t e 1 = √ (log t)n dt 2 t

(n ∈ N0 ).

(37)

0

Both (36) and (37) are recorded by Campbell [208, p. 25]. Making use of 1.1(8), instead of 1.1(1) (or, alternatively, by setting t = log (1/τ ) in (36) and (37)), we obtain 0 (n) (1) =

 Z1   1 n log log dt t

(n ∈ N0 )

(38)

0

and 0

(n)

  Z1   1   1 1 n 1 −2 = log log dt log 2 t t

(n ∈ N0 ).

(39)

0

By using the computations presented here, each of the integral formulas (36), (37), (38) and (39) can be expressed explicitly in terms of γ and ζ (n), at least for n = 1, . . . , 10. Differentiating both sides of 1.1(46) n times, by employing the Leibniz rule for differentiating the 0-product and setting p = 21 in the resulting equation, we obtain Z∞ 0

(log t)n √ dt := I(n) (1 + t) t       n X n (n−k) 1 1 = (−1)k 0 0 (k) k 2 2 k=0

(40) (n ∈ N0 ).

374

Zeta and q-Zeta Functions and Associated Series and Integrals

It is interesting to observe that Z∞ 0

(log t)2n+1 √ dt = 0 (1 + t) t

(n ∈ N0 ),

(41)

which can also be shown as follows, by separating the sum on the right-hand side of (40) into two parts:         n 2n+1 X X  (−1)k 2n + 1 0 (2n+1−k) 1 0 (k) 1 I(2n + 1) =  + 2 2 k k=0

=

n X

k=n+1

     1 2n + 1 (2n+1−k) 1 0 0 (k) k 2 2 k=0       n X (n−k) 1 n+1−k 2n + 1 (n+1+k) 1 0 + (−1) 0 , n−k 2 2 (−1)k



k=0

which, upon reversing the order of the latter sum, yields       n n oX 2n + 1 (2n+1−k) 1 1 I(2n + 1) = 1 + (−1)2n+1 (−1)k 0 0 (k) k 2 2 k=0

= 0.

(42)

Conversely, in the case of even integers, we have Z∞ I(2n) = 0

(log t)2n √ dt (1 + t) t

n−1 X

      2n (2n−k) 1 (k) 1 =2 (−1) 0 0 k 2 2 k=0    2 2n 1 + (−1)n 0 (n) (n ∈ N0 ), n 2 k

(43)

which, in view of the results already presented here, readily yields the following special cases: Z∞ 0

Z∞ 0

(log t)2 √ dt = π 3 , (1 + t) t

(44)

(log t)4 √ dt = 5 π 5 , (1 + t) t

(45)

Series Involving Zeta Functions

Z∞ 0

Z∞ 0

375

(log t)6 √ dt = 61 π 7 , (1 + t) t

(46)

(log t)8 √ dt = 1385 π 9 , (1 + t) t

(47)

(log t)10 √ dt = 50521 π 11 . (1 + t) t

(48)

and Z∞ 0

We conclude by remarking that the evaluations given here may find further applications involving (for example) computation and evaluation of several families of definite integrals.

3.7 Applications of Series Involving the Zeta Function The Multiple Gamma Functions The theory of multiple Gamma functions was applied in several earlier works to evaluate some families of series involving the Riemann Zeta function, as well as to compute the determinants of the Laplacians (see Section 3.4 and Chapter 5). Recently, Choi et al. [269] addressed the converse problem and applied various (known or new) formulas for series associated with the Zeta and related functions with a view to developing the corresponding theory of multiple Gamma functions. We begin by mainly using 1.4(86) to derive the explicit forms of 0n (z) like 1.4(3) and 1.4(91). For example, we have   04 (1 + z) = exp d1 z + d2 z2 + d3 z3 + d4 z4 (   ) (1) ∞  k+2 Y z −( 3 ) k+2 z z2 z3 z4 · 1+ exp − + − , 3 k k 2k2 3k3 4k4 k=1

where 1 1 1 γ 1 1 7 − log A − log B − log(2π), d2 = − + + log(2π) + log A, 24 2 6 144 6 4 2 2 γ 1 π2 11 γ π 2 ζ (3) d3 = − − − log(2π) − , d4 = + + + ; 9 6 12 54 144 24 48 12   05 (1 + z) = exp e1 z + e2 z2 + e3 z3 + e4 z4 + e5 z5 (   ) (2) ∞  k+3 Y z −( 4 ) k+3 z z2 z3 z4 z5 · 1+ exp − + − + , k 4 k 2k2 3k3 4k4 5k5 d1 =

k=1

376

Zeta and q-Zeta Functions and Associated Series and Integrals

where 1 11 0 3 ζ (3) 1 0 1 1 469 − log(2π) + ζ (−1) − + ζ (−3) + ζ (4) − ζ (5), 12 20 20 26 · 32 · 5 8 16 π 2 6 γ 11 3 ζ (3) e2 = + log(2π) − ζ 0 (−1) + , 8 48 4 16 π 2 161 11 1 1 1 e3 = − 5 3 − 3 2 γ − log(2π) + ζ 0 (−1) − ζ (2), 8 6 12 2 ·3 2 ·3 7 1 1 11 1 e4 = + γ+ log(2π) + ζ (2) + ζ (3), 64 16 48 96 16 5 1 1 11 1 e5 = − 5 2 − 3 γ− ζ (2) − ζ (3) − ζ (4). 20 120 20 2 ·3 2 ·3·5 e1 =

By virtue of 1.4(82), to get the constants dj 0 s and ej 0 s involved in (1) and (2), respectively, it is indispensable to evaluate An (1) explicitly. To do this, using 1.4(83) and 1.4(8.6), we find that # " n ∞   X z n + k − 2 X (−1)n−`  z ` n An (z) = + (−1) log 1 + n−1 ` k k `=1 k=1   ∞ ∞ X (−1)n+`−1 ` X n + k − 2 1 z , = ` n−1 k` `=n+1

k=1

which, by means of the expansion: 

 n−1 X n+k−2 1 = (−1)n−1+j s(n − 1, j) kj , n−1 (n − 1)! j=0

immediately yields An (z) =

n−1 ∞ X X (−1)` 1 (−1)j s(n − 1, j) ζ (` − j)z` , (n − 1)! `

(3)

`=n+1

j=0

where s(n, k) denote the Stirling numbers of the first kind given in Section 1.5. Now, we can apply 3.2(63) to (3), and we get    j n−1 X X 1 j 0 An (z) = s(n − 1, j)  (−1)k ζ (−k, 1 + z) zj−k (n − 1)! k j=0

+ (−1)j+1 ζ 0 (−j) −

k=0

j−1 X

(−1)`

`=0

 n−j X ζ (k) − (−1)k zk+j  , k+j k=2


ζ (−`) j−` zj+1 z + Hj + γ j−` j+1

(4)

Series Involving Zeta Functions

377

the special case z = 1 of which is given here for convenience:    j−1 n−1 X X 1 j 0 An (1) = s(n − 1, j)  (−1)k ζ (−k) (n − 1)! k j=0

−

j−1 X

(−1)k

k=0

j−k

k=0

ζ (−k) +

1 Hj + 2γ − j+1


n−j X (−1)k k=2

k+j

(5)

 ζ (k)

(n ∈ N).

By using various identities given in this and the foregoing chapters, some special cases of (5) are explicitly obtained here as follows: 1 γ π2 γ 1 − log(2π), A3 (1) = − − + + log A, 2 2 4 12 36 5 γ π2 ζ (3) 1 1 A4 (1) = + + + − log A − log B, 36 24 432 12 2 2 (6) 37 19 3 7 A5 (1) = − − γ+ ζ (2) + ζ (3) 540 720 160 240 1 1 0 ζ (3) 1 0 + ζ (5) − ζ (−1) + − ζ (−3). 20 3 8π2 6 A1 (1) = γ ,

A2 (1) = 1 +

Next, by taking logarithms on each side of 1.4(3), 1.4(91) and (1), we obtain ∞ X 1 1 1 (−1)n z − z log(2π) + (γ + 1) z2 − ζ (n) zn+1 , 2 2 2 n+1 n=2     3 1 γ 1 1 2 log 03 (1 + z) = − log(2π) − log A z + + log(2π) + z 8 4 4 4 8   ∞ ∞ 1 γ 3 1 X (−1)n 1 X (−1)n − + z − ζ (n)zn+1 + ζ (n)zn+2 , 4 6 2 n+1 2 n+2

log 02 (1 + z) =

n=2

n=2

and 

 7 1 1 log 04 (1 + z) = − log A − log B − log(2π) z 24 2 6   1 1 γ 1 + − + + log(2π) + log A z2 2 72 3 2   1 4 1 − + γ + log(2π) z3 6 3 2   ∞ 1 11 1 X (−1)n + + γ z4 − ζ (n)zn+3 24 6 6 n+3 n=2

+

1 2

∞ X

(−1)n

n=2

n+2

∞

ζ (n)zn+2 −

1 X (−1)n ζ (n)zn+1 , 3 n+1 n=2

378

Zeta and q-Zeta Functions and Associated Series and Integrals

which, upon employing the special case a = 1 of 3.2(63), can be expressed in terms of the Zeta functions as follows: 1 + log A − z log 0(1 + z) + ζ 0 (−1, 1 + z), 12   1 ζ (3) 1 1 − log A zn + log 03 (1 + z) = − + log A + 24 2 12 8π 2   1 1 1 + (z2 − z) log 0(1 + z) + − z ζ 0 (−1, 1 + z) + ζ 0 (−2, 1 + z), 2 2 2 log 02 (1 + z) = −

(7)

(8)

and   1 1 ζ (3) 1 0 ζ (3) 1 z − + log A + ζ (−3) + − log A − 36 3 12 8π 2 6 8π 2   1 1 1 + − + log A z2 − (z3 − 3z2 + 2z) log 0(1 + z) (9) 24 2 6   1 1 1 2 z −z+ ζ 0 (−1, 1 + z) + ζ 0 (−3, 1 + z), + 2 3 6

log 04 (1 + z) = −

where we have also made use of 7.1(24) and  the 1known identity 2.3(22). 1 Analogous to the classical result 0 2 = π 2 , setting z = − 12 in (7), (8) and (9), we obtain 02

 

1

1

1

3

= 2− 24 · π 4 · e− 8 · A 2 ,     1 3 3 1 7ζ (3) − 24 1 16 · π · exp − + ·A2 , 03 2 = 2 8 32 π 2 1 2

(10) (11)

and 04

  1 2

229

5

265

23

3

5

= 2− 5760 · π 32 · e− 2304 · A 16 · B 4 · C 16 .

(12)

It is noted that (10) and (11) are the same as 1.4(18) and 1.4(97), respectively. By integrating the very three equations before (7) from 0 to z and using the special case a = 1 of 3.2(63), we get Zz

log 02 (1 + t) dt =

  1 1 1 ζ (3) 0 − + ζ (−1) z − z2 log(2π) + z3 12 4 6 4π 2

0

− z ζ 0 (−1, 1 + z) + ζ 0 (−2, 1 + z),

(13)

Series Involving Zeta Functions

Zz

log 03 (1 + t) dt =

379

  ζ (3) 1 0 1 ζ (3) 1 − − ζ (−3) + + log A z 8π 2 2 8π 2 12 2

0

 1 1 1 1 4 z 0 1 − log(2π) − log A z2 + [1 + log(2π)] z3 − z − ζ (−1, 1 + z) + 24 8 2 12 24 2 

+

1 2 0 1 1 z ζ (−1, 1 + z) + ζ 0 (−2, 1 + z) − zζ 0 (−2, 1 + z) + ζ 0 (−3, 1 + z), 2 2 2

(14)

and Zz

log 04 (1 + t) dt

0

  1 0 ζ (3) ζ (5) 19 1 0 ζ (3) 1 0 − − + ζ (−3) − + ζ (−1) − ζ (−3) z 720 3 12π 2 2 8π4 8π2 6     ζ (3) 1 1 1 0 1 0 1 2 ζ (−1) − log(2π) z + + log(2π) − ζ (−1) z3 − + 2 18 12 6 16 π 2 12 (15)    1 1 1 5 13 − 1 + log(2π) z4 + z − z − 3z2 + 2z ζ 0 (−1, 1 + z) 24 2 120 6   1 0 1 1 1 2 + z −z+ ζ (−2, 1 + z) − (z − 1)ζ 0 (−3, 1 + z) + ζ 0 (−4, 1 + z). 2 3 2 6 =

Upon setting z = 1 and z = recorded identities, we have Z1

log 02 (1 + t) dt = −

1 2

in (13), (14) and (15) and using some previously

1 1 − log(2π) + 2 log A, 12 4

(16)

0

Z1

log 03 (1 + t) dt =

3ζ (3) 1 log(2π), − 2 24 8π

(17)

log 04 (1 + t) dt =

13 3ζ (3) 1 2 + − log(2π) + log C, 2 2160 16 π 48 3

(18)

0

Z1 0 1

Z2 0

log 02 (1 + t) dt = −

1 1 1 7ζ (3) (1 + log 2) − log π + log A + , 24 16 4 16 π 2

(19)

380

Zeta and q-Zeta Functions and Associated Series and Integrals 1

Z2

log 03 (1 + t) dt = −

29 1 1 − log 2 − log π 256 1920 48

0

1 3 15 + log A + log B + log C, 16 4 16

(20)

and 1

Z2

log 04 (1 + t) dt =

73 17 3 1 − log 2 − log π + log A 69120 1920 256 32

0

+

(21)

77 47ζ (3) 31ζ (5) − . log C + 96 384π 2 128 π 4

Barnes [94, p. 283] expressed log 02 (z + a) as an integral of log 0(t + a) as follows: Zz 1 z2 log 0(t + a) dt = [log(2π) + 1 − 2a] z − + (z + a − 1) log 0(z + a) 2 2 0 (22) + log 02 (z + a) + (1 − a) log 0(a) − log 02 (a). Similarly, we have   Zz a2 1 1 − (a − 1) log(2π) + 2 log A + −a z log 02 (t + a)dt = 4 2 2 0

1 1 + [2a − 2 − log(2π)]z2 + z3 + (z + a − 2) log 02 (z + a) 4 6 + 2 log 03 (z + a) + (2 − a) log 02 (a) − 2 log 03 (a),

Zz

(23)

log 03 (t + a) dt

0

 3 2 1 3 3 1 1 5 − a + a − a + (3 − 2a) log A + log B + (a − 1)(a − 2) log(2π) z = 8 6 4 6 2 4       1 2 3 5 1 3 1 1 1 2 + − a + a− + a− log(2π) − log A z + − a+ log(2π) z3 4 4 12 4 8 4 6 12 1 4 − z + (z + a − 3) log 03 (z + a) + (3 − a) log 03 (a) + 3 log 04 (z + a) − 3 log 04 (a), 24 (24) 

and Zz

log 04 (t + a) dt = β1 z + β2 z2 + β3 z3 + β4 z4 + β5 z5 − 4 log 05 (a)

0

+ (z + a − 4) log 04 (z + a) + (4 − a) log 04 (a) + 4 log 05 (z + a),

(25)

Series Involving Zeta Functions

381

where 11 2 1 3 1 4 1937 19 + γ −a+ a − a + a 2160 90 12 3 24    1  3 − a − 6 a2 + 11 a − 6 log(2 π) − a2 − 4a + 1 ζ 0 (−1) 12 ζ (3) 2 3 7 1 1 1 ζ (2) − ζ (3) − ζ (4) − ζ (5), + (1 − 3a) 2 + ζ 0 (−3) − 8 3 20 30 5 5 π   1 11 1 2 1 3 1 2 1 11 β2 = − + a− a + a − a − a+ log(2 π) 2 12 2 12 8 2 24 3 ζ (3) + (2 − a) ζ 0 (−1) − , 16 π 2 11 1 1 2 1 1 β3 = − a+ a + (2 − a) log(2 π) − ζ 0 (−1), 36 3 12 12 3 1 1 1 1 a− log(2 π), and β5 = . β4 = − + 12 24 48 120 β1 =

To derive (25), for example, by setting z = t + a in 1.1(2) and z = t + a − 1 in 1.4(3), 1.4(91), (1) and (2) and taking the logarithmic derivatives on each side of the resulting equations, we find that  ∞  X 0 0 (t + a) 1 1 = −γ − − , 0(t + a) t+a−1+k k k=1

020 (t + a) 1 1 = − log(2π) + (1 + γ )(t + a − 1) 02 (t + a) 2 2   ∞ X 1 1 t+a−1 + k − + − , t+a−1+k k k2 k=1

030 (t + a) = c1 + 2 c2 (t + a − 1) + 3 c3 (t + a − 1)2 03 (t + a)   ∞ X 1 1 1 t + a − 1 (t + a − 1)2 + k(k + 1) − + − + , 2 t + a−1 + k k k2 k3 k=1

and 040 (t + a) = d1 + 2 d2 (t + a − 1) + 3 d3 (t + a − 1)2 + 4 d4 (t + a − 1)3 04 (t + a)  ∞  X 1 t+a−1 k+2 1 + − + − 3 t+a−1+k k k2 k=1  (t + a − 1)2 (t + a − 1)3 − , + k3 k4

382

Zeta and q-Zeta Functions and Associated Series and Integrals

all of which can be applied in conjunction with 050 (t + a) = e1 + 2 e2 (t + a − 1) + 3 e3 (t + a − 1)2 + 4 e4 (t + a − 1)3 05 (t + a)  ∞  X k+3 1 1 t+a−1 4 − + − + 5 e5 (t + a − 1) + 4 t+a−1+k k k2 k=1  (t + a − 1)2 (t + a − 1)3 (t + a − 1)4 + − + k5 k3 k4 to obtain 050 (t + a) = e1 + ω1 (t + a − 1) + ω2 (t + a − 1)2 + ω3 (t + a − 1)3 05 (t + a) 0 0 (t + a) 0 0 (t + a) 1 1 + ω4 (t + a − 1)4 − (t + a − 1) − (t + a − 1) 2 4 0(t + a) 4 02 (t + a) 030 (t + a) 1 040 (t + a) 1 − (t + a − 1) , − (t + a − 1) 4 03 (t + a) 4 04 (t + a)

(26)

where 1 11 3 ζ (3) 1 11 1 + log(2π) − ζ 0 (−1) + , ω2 = − − log(2π) + ζ 0 (−1), 4 48 48 8 4 32 π 2 1 1 1 ω3 = + log(2π), and ω4 = − . 12 48 96 ω1 =

Finally, upon integrating both sides of (26) from t = 0 to t = z and using (22), (23) and (24), we are led to the desired identity (25). The foregoing methods can also be applied with a view to deriving explicit expressions for the multiple Gamma functions 0n of higher order and various other related identities.

Mathieu Series Almost twelve decades ago, Mathieu investigated an interesting series S(r) in the study of elasticity of solid bodies. Since then, many authors have studied various problems arising from the Mathieu series S(r) in various diverse ways. Here, we present a relationship between the Mathieu series S(r) and certain series involving the Zeta functions (see [303]). By means of this relationship, we then express the Mathieu series S(r) in terms of the Trigamma function ψ 0 (z) or (equivalently) the Hurwitz (or generalized) Zeta function ζ (s, a). Accordingly, various interesting properties of S(r) can be obtained from those of ψ 0 (z) and ζ (s, a). Among other results, certain integral representations of S(r) are deduced here, by using the aforementioned relationships among S(r), ψ 0 (z) and ζ (s, a).

Series Involving Zeta Functions

383

´ Emile Leonard Mathieu (1835–1890) [803] investigated the following infinite series: S(r) =

∞ X

2n

n=1


2 n2 + r2

r ∈ R+




(27)

in the study of elasticity of solid bodies (see also [418]), R+ being (as usual) the set of positive real numbers. Pog´any et al. [902] introduced an alternating version of the Mathieu series (1.1) as follows: ˜ = S(r)

∞ X

(−1)n−1

n=1

2n n2 + r2


r ∈ R+ .


2

(28)

Since the time of Mathieu, many authors (see, e.g., [30], [376], [403], [407], [418], [897], [898], [899], [900], [901], [902], [903], [904], [925], [1110], [1155], [1156], [1157], [1158], [1159], [1160] and [1161]; see also the references cited in each of these works) have investigated various problems arising from the Mathieu series (27) and its extensions and generalizations in various diverse ways. In particular, Poga´ ny et al. [902] presented the following integral representations of the Mathieu series (27) and the alternating Mathieu series (28): 1 S(r) = r

Z∞ 0

t sin(rt) dt et − 1

(29)

t sin(rt) dt. et + 1

(30)

and ˜ =1 S(r) r

Z∞ 0

Srivastava and Tomovski [1110], conversely, defined the following five-parameter family of generalized Mathieu series: (α,β) (α,β) (r; a) = Sµ (r; {ak }) = Sµ

∞ X n=1

β

2 an

aαn + r2


µ

where it is tacitly assumed that the positive sequence   ∞ a := {ak }k=1 lim ak = ∞ k→∞


r, α, β, µ ∈ R+ ,

(31)

(32)

is so chosen (and then the positive parameters α, β and µ) are so constrained that the infinite series in (31) converges, that is, that the following auxiliary series: ∞ X

1

µα−β n=1 an

is convergent.

384

Zeta and q-Zeta Functions and Associated Series and Integrals

Several authors have considered some interesting variants and special cases of the generalized Mathieu series (31) (see, e.g., [897], [899], [902] and [925]). And, very recently, Elezovic´ et al. [407] derived several results involving the Laplace, Fourier and Mellin transforms of various functions belonging to the family of generalized ˜ Mathieu series (31), including, for example, the Laplace transforms of S(r) and S(r) as given below:


L S(r) (x) =

Z∞  0

t et − 1



t t e +1



  t dt x

(33)

  t arctan dt. x

(34)

arctan

and
˜ L S(r) (x) =

Z∞  0

The Dirichlet Eta function (or the alternating Riemann Zeta function) η(s) is defined by η(s) =

∞ X (−1)n−1 ns


<(s) > 0 .

(35)

n=1

It is easy to find from 2.3(1) and (35) that   η(s) = 1 − 21−s ζ (s)


<(s) > 1 .

(36)

We express the Mathieu series in (27) and the alternating Mathieu series in (28) as sums involving the Riemann Zeta function, which can also be evaluated in terms of the Trigamma function ψ 0 (z) and so the Hurwitz Zeta function ζ (s, a) as follows: Each of the following series representations holds true:

S(r) = 2

∞ X

(−1)k−1 k ζ (2k + 1) r2(k−1)

(|r| < 1)

(37)

(−1)k−1 k η(2k + 1) r2(k−1)

(|r| < 1),

(38)

k=1

and ˜ =2 S(r)

∞ X k=1

where ζ (s) and η(s) are the Riemann Zeta function and the Dirichlet Eta function given in 2.3(1) and (35), respectively.

Series Involving Zeta Functions

385

Proof. By applying the following elementary identity: ∞

X 1 = (−1)k (k + 1) zk 2 (1 + z)

(|z| < 1)

(39)

k=0

to (27), we obtain ∞ X 1 2n S(r) =  n4 1 + (r/n)2 2 n=1

=

∞ ∞ X r2k 2n X (−1)k (k + 1) 2k 4 n n n=1 ∞ X

=2

k=0

(−1)k (k + 1) r2k

=2

∞ X n=1

k=0 ∞ X

 r  < 1; n ∈ N n 1

n2k+3

(−1)k (k + 1) ζ (2k + 3) r2k ,

k=0

where the rearrangement of the sums in the penultimate step is guaranteed by the absolute convergence of the double series involved. Finally, upon replacing the summation index k by k − 1 in the last expression, we are led to the assertion (37). Similarly, we can prove the assertion (38). By using (37) and (38), we obtain: Each of the following relationships holds true:  i  0 ψ (1 + ir) − ψ 0 (1 − ir) 2r i [ζ (2, 1 + ir) − ζ (2, 1 − ir)] S(r) = 2r

S(r) =

 √  0 < |r| < 1; i = −1 ,  √  0 < |r| < 1; i = −1

(40) (41)

and   ˜ = S(r) − 1 S r S(r) 4 2

(|r| < 1),

(42)

where ψ 0 (z) and ζ (s, a) are the Trigamma function and the Hurwitz (or generalized) Zeta function given in 1.3(52) and 2.1(1), respectively.  Proof. We recall the following known identity (see [1094, p. 160, Eq. (16)]): ∞ X k=1

1 ζ (2k + 1) t2k = − [ψ(1 + t) + ψ(1 − t)] − γ 2

(|t| < 1),

(43)

where γ denotes the Euler-Mascheroni constant given in 1.1(3) (also see Section 1.2).

386

Zeta and q-Zeta Functions and Associated Series and Integrals

Differentiating each side of (43) with respect to t and dividing the resulting identity by t, we get 2

∞ X

k ζ (2k + 1) t2(k−1) = −

k=1

 1  0 ψ (1 + t) − ψ 0 (1 − t) 2t

(0 < |t| < 1). (44)

√
By setting t = i r i = −1 in (44), and making use of (37) we prove the assertion (40). Similarly, by applying 1.3(53) to (40), we obtain (41). Furthermore, by combining (37), (38) and (36), we can prove (42). It is noted that, in view of the relationship in (42), several properties of the alter˜ nating Mathieu series S(r) can easily be obtained from the corresponding properties of the Mathieu series S(r). First of all, we analytically continue the Mathieu series S(r) from r ∈ R+ to the complex plane C as follows: The Mathieu series S(r) is analytic on the punctured complex plane n C \ z : z = ±n i

 √ o n ∈ N; i = −1 . 

Proof. It is obvious that each term 2n/(n2 + r2 ) in the series (1.1) defining the S(r) is analytic on n C \ z : z = ±n i

 √ o n ∈ N; i = −1 .

Since the series for S(r) is seen to be uniformly convergent on any compact subset D in n  √ o C \ z : z = ±n i n ∈ N; i = −1 , S(r) is analytic on D. Now, by the well-known principle due to Weierstrass (see, e.g., [1025, p. 88, Theorem]), the Mathieu series S(r) is analytic on n C \ z : z = ±n i

 √ o n ∈ N; i = −1 .

By means of the relationships given in (40), (41) and (42) and by making use of several known integral representations of ζ (s, a) and ψ(z), for example, 2.2(21), 2.2(25), 1.3(18), 1.3(24) and another known integral formula for ψ(z) (see [1225, p. 261, Example 19]):

ψ(z) = log z −

Z1 0

1 − t + log t z−1 t dt (1 − t) log t


<(z) > 0 ,

(45)

Series Involving Zeta Functions

387

˜ we present various integral representations of S(r) and S(r), including (29) and (30) (see [303]). Each of the following representations holds true:  n  1 i X 1 1 + − Bk S(r) =
2 + 1 + r2 2r (1 + ir)k+1 (1 − ir)k+1 1 + r2 k=2 ! Z∞ n X 1 1 Bk k−1 + − t t e−t sin(rt) dt r et − 1 k! k=0 0  √  <(r) > 0; n ∈ N0 ; i = −1 , 1

(46)

where Bk denotes the Bernoulli numbers and an empty sum is (as usual) understood to be nil; 1 S(r) = r

Z1 0

S(r) =

1 r

Z∞ 0

log t sin(r log t) dt 1−t

log(1 + t) sin [r log(1 + t)] dt t(1 + t)

1 1 + S(r) = 2 r 1+r 1


<(r) > 0 ;

Z1 0

(47)


<(r) > 0 ;

1 − t + log t sin(r log t) dt 1−t

1 S(r) = +4
2 + 1 + r2 1 + r2

Z∞ 0


<(r) > 0 ;

1 + r2 − t2 t dt
2 2π t 1 + t2 − r2 − 4 r2 e − 1

(48)

(49)


<(r) > 0 . (50) 

Proof. By applying 2.2(25) and the third integral representation in 2.2(21) to (41), we obtain (46) and (47), respectively. Differentiating the right-hand side of each of the integral representations 1.3(18), (45) and 1.3(24) with respect to z under the sign of integration (which can be validated by means of a known result [1225, p. 74, Corollary]) and applying each of the resulting integral representations of the Trigamma function ψ 0 (z) to (40), we obtain (48) to (49), respectively. It is noted that formula (29) can be derived by employing the first integral representation of 2.2(21) in the identity (41) and the application of (29) to the relationship (42) yields (30). It is also noted that, even though the identities (46) to (50) are proven under the condition that 0 < |r| < 1, (in view of observation above) each of these identities holds true, by the principle of analytic continuation, throughout the extended region given by <(r) > 0.

388

Zeta and q-Zeta Functions and Associated Series and Integrals

As a by-product of the main identities (46) to (50), we obtain a variety of interesting identities, some of which are stated below. Each of the following results holds true: 1 1 + S(r) =
2 + 2 2 r 1 + r 1+r 1

Z∞  0

1 1 1 − + et − 1 t 2



t e−t sin(rt) dt

(51)


<(r) > 0 , S(r) =

  1 1 + O 2 1+r r4

(r → ∞; r ∈ R+ )

(52)

and ˜ = 1 S(r) 2r

Z1 0

r h r  i log t sin log t 4 cos log t − 1 dt 1−t 2 2


<(r) > 0 .

(53) 

Proof. The equation (51) is a special case of (46) when n = 1. By making use of the generating function of Bernoulli numbers (see 1.7(2) and 1.7(7)): ∞

t t X t2k − 1 + = B 2k et − 1 2 (2k)!

(|t| < 2π)

(54)

k=1

and the following entry in the readily available tables of Fourier sine transforms (see, e.g., [421, p. 72, Entry 2.4(3)]): Z∞

2 −t

t e

 sin(r t) dt = 2r

0

1 1 + r2

3   3 − r2

(r > 0),

(55)

in conjunction with (51), we obtain (52). Furthermore, if the various alreadydeveloped integral representations of S(r) are applied to the relationship (42), several ˜ can be obtained. For example, by applying (47) to (42), integral representations of S(r) we obtain the last identity (53). Each of the following identities holds true: T(r) :=

∞ X n=1

8n n2 + r2


3

i  i  0 1 h (2) 0 (2) ψ (1 + ir) − ψ (1 − ir) + ψ (1 + ir) + ψ (1 − ir) (56) 2 r3 2 r2 i 1 = 3 [ζ (2, 1 + ir) − ζ (2, 1 − ir)] − 2 [ζ (3, 1 + ir) + ζ (3, 1 − ir)] 2r r √
0 < |r| < 1; i = −1

=

Series Involving Zeta Functions

389

and
 Z∞  2 3 + r2 1 1 1 1 − + T(r) =
3 + 3 et − 1 t 2 r 1 + r2 0

· t e−t [sin(rt) − rt cos(rt)] dt


<(r) > 0 .

(57) 

Proof. Upon differentiating (27) and (40) with respect to r, if we combine the resulting identities and make use of 1.3(53), we find a new series T(r), which is expressed in terms of the Polygamma functions and the Hurwitz Zeta function as in (56). By applying (51) to (56), we obtain (57). 

Problems 1. Evaluate the following special values of the G-function given in Section 1.4:

G

 

√  1 √ n  o− 2 3 ζ 2, 3 1 4 1 π 3 3  · 3 72 · A− 3 · 0 1 = exp  + − ; 3 9 54 36π

G

 

√  1 √ n  o 1 3 ζ 2, 3 1 13 1 4 1 π 3  · 2− 3 · 3 72 · π − 3 · A− 3 · 0 1 3 ; = exp  − + 3 9 54 36π



1 3



2 3

√  1 n  o− 5 3 ζ 2, 3 19 61 5 5 5 π 3 3  · 2 72 · 3− 144 · π 12 · A− 6 · 0 1 − ; G 16 = exp  + 3 72 36 24π  √  1 √ n  o 1   3 ζ 2, 3 17 11 1 5 π 3 5  · 2− 72 · 3 144 π − 4 · A− 6 · 0 1 3 . G 56 = exp  − + 3 72 36 24π 

√

 

(Gosper [498]; Choi and Srivastava [289, 294]) 2. Applying the results of Problem 1, derive the following closed-form evaluations of series involving the Riemann Zeta function: √ √   1 ζ (k) γ π 3 3 (−1) = 1 + + − ζ 2, 6 18 12π 3 (k + 1) · 3k k=2    1 1 1 + log 2− 2 · 3 24 · π − 2 · A−4 · 0 13 ; √ √   ∞ X ζ (k) − 1 11 γ π 3 3 1 (−1)k = + + − ζ 2, 6 6 18 12π 3 (k + 1) · 3k k=2  13 73   1 + log 2− 2 · 3 24 · π − 2 · A−4 · 0 13 ; ∞ X

k

390

Zeta and q-Zeta Functions and Associated Series and Integrals ∞ X k=2

√ √   γ 312π ζ (k) π 3 1 = − + − 2, 6 18 ζ 3 (k + 1) · 3k  n  o−1  13 1 1 ; + log 2 2 · 3− 24 · π 2 · A4 · 0 13

√ √   ∞ X ζ (k) − 1 7 γ π 3 3 1 = − + − ζ 2, 6 6 18 12π 3 (k + 1) · 3k k=2  n  o−1  85 1 7 + log 2 2 · 3− 24 · π 2 · A4 · 0 13 ; √ √   1 π 3 1 1 ζ (2k) 3 = + − ζ 2, − log 3; 2 18 12π 3 4 (2k + 1) · 32k k=1 √ √   ∞  3  X 3 1 ζ (2k) − 1 11 π 3 −2 − 14 + − ζ 2, + log 2 · 3 ; = 6 18 12π 3 (2k + 1) · 32k ∞ X

k=1 ∞ X k=1

 n  o−6  ζ (2k + 1) 3 24 3 − 74 · π · A · 0 31 = −3 − γ + log 2 · 3 ; (k + 1) · 32k

 ∞ n  o−6  X ζ (2k + 1) − 1 30 − 79 4 · π 3 · A24 · 0 1 ; = −2 − γ + log 2 · 3 3 (k + 1) · 32k k=1 √ √     ∞ X ζ (k) 2 k 3 γ π 3 1 (−1)k = 1+ − + ζ 2, k+1 3 3 36 24π 3 k=2  n  o−1  1 1 23 + log 2 2 · 3− 48 · π 2 · A−2 · 0 13 ; ∞ X

(−1)k

k=2

ζ (k) − 1 k+1

√ √   k  2 5 γ π 3 3 1 = + − + ζ 2, 3 3 3 36 24π 3  n  o−1  1 49 3 1 + log 2 2 · 3 48 · 5− 2 · π 2 · A−2 · 0 13 ;

√ √     ∞ X 3 ζ (k) 2 k γ π 3 1 =− − + ζ 2, k+1 3 3 36 24π 3 k=2  1   1 1 + log 2− 2 · 3− 48 · π − 2 · A2 · 0 13 ; √ √     ∞ X ζ (k) − 1 2 k 4 γ π 3 3 1 = − − + ζ 2, k+1 3 3 3 36 24π 3 k=2  1   73 1 + log 2− 2 · 3− 48 · π − 2 · A2 · 0 13 ; √ √     ∞ X ζ (2k) 2 2k 1 π 3 3 1 1 = − + ζ 2, − log 3; 2k + 1 3 2 36 24π 3 4 k=1 √ √     ∞  1  X ζ (2k) − 1 2 2k 3 π 3 3 3 1 = − + ζ 2, + log 3− 4 · 5− 4 ; 2k + 1 3 2 36 24π 3 k=1

Series Involving Zeta Functions

391

   ∞ n  o3  X 3 11 3 3 ζ (2k + 1) 2 2k = − − γ + log 2− 2 · 3 16 · π − 2 · A6 · 0 13 ; k+1 3 2 k=1

   ∞ n  o3  X 61 9 3 3 ζ (2k + 1) − 1 2 2k 1 = − − γ + log 2− 2 · 3− 16 · 5 4 · π − 2 · A6 · 0 13 ; k+1 3 2 k=1

√ √   ζ (k) 1 γ π 3 3 (−1) = 1+ + − ζ 2, 12 6 4π 3 (k + 1) · 6k k=2  n  o2  11 11 + log 2− 12 · 3 24 · π −1 · A−5 · 0 13 ;

∞ X

∞ X

k

(−1)k

k=2

∞ X k=2

√ √   γ π 3 3 1 ζ (k) − 1 23 + + − ζ 2, = 12 12 6 4π 3 (k + 1) · 6k  n  o2  61 155 + log 2 12 · 3 24 · 7−6 · π −1 · A−5 · 0 13 ; √ √   γ ζ (k) π 3 3 1 = − + − ζ 2, 12 6 4π 3 (k + 1) · 6k  n  o−2  11 11 + log 2 12 · 3− 24 · π · A5 · 0 31 ;

√ √   ∞ X ζ (k) − 1 13 3 γ π 3 1 = − + − ζ 2, 12 12 6 4π 3 (k + 1) · 6k k=2  n  o−2  − 155 6 5 − 61 12 24 ·3 · 5 · π · A · 0 13 ; + log 2 ∞ X k=1

√  √  ζ (2k) 1 π 3 3 1 = + − ζ 2, ; 2 6 4π 3 (2k + 1) · 62k

√ √   ∞   X ζ (2k) − 1 3 π 3 3 1 = + − ζ 2, + log 53 · 7−3 ; 2k 2 6 4π 3 (2k + 1) · 6 k=1

 ∞ n  o−24  X ζ (2k + 1) 11 − 11 2 · π 12 · A60 · 0 1 = −6 − γ + log 2 · 3 ; 3 (k + 1) · 62k k=1

∞ X k=1

∞ X k=2

ζ (2k + 1) − 1 = −5 − γ (k + 1) · 62k  n  o−24  155 + log 2−61 · 3− 2 · 536 · 736 · π 12 · A60 · 0 13 ;

ζ (k) (−1) k+1 k

√ √   k  5 5 π 3 3 1 = 1+ γ − + ζ 2, 6 12 30 20π 3  n  o−2  49 61 + log 2 60 · 3− 120 · π · A−1 · 0 13 ;

392

Zeta and q-Zeta Functions and Associated Series and Integrals ∞ X

(−1)

√ √   k  5 19 3 5 π 3 1 = + γ− + ζ 2, k+1 6 12 12 30 20π 3  n  o−2  121 83 6 + log 2 60 · 3 120 · 11− 5 · π · A−1 · 0 31 ;

k ζ (k) − 1

k=2

√ √     ∞ X ζ (k) 5 k 3 5 π 3 1 =− γ − + ζ 2, k+1 6 12 30 20π 3 k=2  n  o2  61 −1 − 49 60 120 ; · 3 · π · A · 0 31 + log 2 √ √     ∞ X ζ (k) − 1 5 k 17 5 π 3 1 3 − γ− + ζ 2, = k+1 6 12 12 30 20π 3 k=2  n  o2  83 121 ; + log 2− 60 · 3− 120 · π −1 · A · 0 13 √ √     ∞ X ζ (2k) 5 2k 1 π 3 3 1 = − + ζ 2, ; 2k + 1 6 2 30 20π 3 k=1

∞ X k=1

ζ (2k) − 1 2k + 1

√ √   2k  5 3 π 3 3 1 3 = − + ζ 2, − log 11; 6 2 30 20π 3 5

   ∞ n  o 24  X 61 12 12 49 ζ (2k + 1) 5 2k 6 5 = − − γ + log 2− 25 · 3 50 · π − 5 · A 5 · 0 13 ; k+1 6 5 k=1

∞ X k=1

∞ X k=2

ζ (2k + 1) − 1 k+1

 2k 5 1 = − −γ 6 5  n  o 24  83 12 121 36 12 5 ; + log 2− 25 · 3− 50 · 11 25 · π − 5 · A 5 · 0 13

√ √   k  3 4 19 2 π 3 1 = + γ+ − ζ 2, 3 12 3 72 48π 3  1 1   3 1 + log 2− 2 · 3 96 · 7− 4 · π − 2 · A−1 · 0 13 ; √ √     ∞ X ζ (k) − 1 4 k 17 2 3 π 3 1 = − γ+ − ζ 2, k+1 3 12 3 72 48π 3 k=2   n  o −1 1 1 − 49 1 2 96 2 + log 2 · 3 ·π ·A· 0 3 ;

(−1)k

ζ (k) − 1 k+1

√ √     ∞  1  X 3 ζ (2k) − 1 4 2k 3 π 3 3 1 = + − ζ 2, + log 3− 4 · 7− 8 ; 2k + 1 3 2 72 48π 3 k=1

   ∞ n  o− 3  X 3 25 9 3 3 ζ (2k + 1) − 1 4 2k 1 2 = − − γ + log 2 4 · 3− 64 · 7 16 · π 4 · A 2 · 0 13 ; k+1 3 8 k=1

Series Involving Zeta Functions ∞ X

(−1)

k=2

393

√ √   k  5 47 5 3 π 3 1 = + γ− + ζ 2, k+1 3 30 6 90 60π 3  n  o−1  7 59 1 4 + log 2− 10 · 3− 120 · π 2 · A− 5 · 0 13 ;

k ζ (k) − 1

√ √     ∞ X ζ (k) − 1 5 k 43 5 3 π 3 1 = − γ− + ζ 2, k+1 3 30 6 90 60π 3 k=2    1 4 1 1 + log 2− 2 · 3− 120 · π − 2 · A 5 · 0 13 ; √ √     ∞   3 X 1 ζ (2k) − 1 5 2k 3 π 3 3 1 = − + ζ 2, + log 2− 5 · 3− 4 ; 2k + 1 3 2 90 60π 3 k=1

   ∞ n  o 6  X 29 3 24 3 ζ (2k + 1) − 1 5 2k 2 5 = − − γ + log 2 25 · 3 100 · π − 5 · A 25 · 0 13 ; k+1 3 25 k=1

∞ X

(−1)

k=2

√ √   k  7 131 7 π 3 1 3 = + γ+ − ζ 2, k+1 6 84 12 42 28π 3  n  o2  83 − 71 − 67 −1 − 57 84 168 + log 2 · 3 · 13 · π · A · 0 13 ;

k ζ (k) − 1

√ √     ∞ X 7 π 3 1 ζ (k) − 1 7 k 121 3 = − γ+ − ζ 2, k+1 6 84 12 42 28π 3 k=2   n  o −2 83 5 71 ; + log 2 84 · 3− 168 · π · A 7 · 0 31 √ √     ∞ X ζ (2k) − 1 7 2k 3 π 3 1 3 3 − ζ 2, = + − log 13; 2k + 1 6 2 42 28π 3 7 k=1

∞ X k=1 ∞ X

ζ (2k + 1) − 1 k+1

(−1)k

k=2

ζ (k) − 1 k+1

 2k  n  o− 24  71 83 12 60 7 5 7 = − − γ + log 2 49 · 3− 98 · π 7 · A 49 · 0 13 ; 6 49 

11 6

k =

√ √   203 11 π 3 3 1 + γ− + ζ 2, 132 12 66 44π 3  n  o−2  133 6 5 6 109 ; + log 2 132 · 3− 264 · 5 11 · 17− 11 · π · A− 11 · 0 31

√ √     ∞ X ζ (k) − 1 11 k 193 11 π 3 3 1 = − γ− + ζ 2, k+1 6 132 12 66 44π 3 k=2  n  o2  133 5 − 109 −1 132 264 11 + log 2 · 3 · π · A · 0 31 ; √ √     ∞  3  X 3 ζ (2k) − 1 11 2k 3 π 3 3 1 = − + ζ 2, + log 5 11 · 17− 11 ; 2k + 1 6 2 66 44π 3 k=1

394

Zeta and q-Zeta Functions and Associated Series and Integrals

  ∞ X ζ (2k+1)−1 11 2k k=1

k+1

6

5 −γ 121  n  o 24  133 36 36 12 60 109 11 . + log 2− 121 · 3 242 · 5− 121 · 17 121 · π − 11 · A 121 · 0 13

=−

(Choi and Srivastava [289, 294]) 3. Prove that ∞ λ   X ζ (m, a) m+λ X λ 0 z = ζ (−k, a − z) zλ−k − ζ 0 (−λ, a) m+λ k

m=2

−

k=0

λ−1 X `=0

ζ (−`, a) λ−` zλ+1 z − [ψ(λ + 1) − ψ(a) + γ ] λ−` λ+1

(λ ∈ N0 ; |z| < |a|). (Kanemitsu et al. [628])

4. Prove the following identity: 1

∞ X k=2

γ 1 5 1 4 (−1)k ζ (k) = + + log 2 + log π − (k + 1)(k + 2) 6 2 18 3 3

Z2

log 0(t) dt,

0

where γ denotes the Euler-Mascheroni constant defined by 1.1(2). (Janos and Srivastava [606]) 5. Prove the following companion of the summation formulas deduced in Problems 46 and 47 (Chapter 1): ζ (k − 1) =

∞ X n=k

(−1)n+k s(n, k). (n − 1)(n − 2) · (n − 1)!

(cf. Jordan [614, p. 339]; see also Equation 3.5(16) and Hansen [531, p. 348]) 6. For 8(z, s, a) defined by 2.5(1), derive the following generalization of the expansion formula 3.2(7): ∞ X (λ)n 8(z, λ + n, a) tn = 8(z, λ, a − t) n!

(|t| < |a|; λ 6= 1).

n=0

(cf. Equation 2.5(33)) 7. For every nonnegative integer `, prove the following formula: ∞ X

8n (z, k, a)

k=2

−

`   X tk+` ` 0 = 8 (z, −k, a − k)t`−k k+` k n k=0

`−1 X k=0

8n (z, −k, a)

t`−k

  t`+1 − H` Ln,1 (z, a) + Ln,2 (z, a) − 80n (z, −`, a) `−k `+1 (|t| < a ; |z| < 1 ; ` ∈ N0 ),

Series Involving Zeta Functions

395

where Ln,1 (z, a) and Ln,2 (z, a) are given, respectively, as Ln,1 (z, a) := lim {(s + `)8n (z, s + ` + 1, a)} s→−`

and Ln,2 (z, a) := lim {8n (z, s + ` + 1, a) + (s + l)80n (z, s + ` + 1, a)}. s→−`

(See Choi et al. [273, Theorem 2]) 8. Show that ∞ X

(−1)n−1


2 Hn−1 + Hn2 On n

n=1

=

π4 π2 7 + log2 2 − log 2 ζ (3). 96 8 4 (See Zheng [1264])

9. Show that, for λ, µ ∈ N0 ,   n  X n + µn n + λn

Hλn+k n−k  
2n + λn + µn = Hλn+n + Hλn+µn+n − Hλn+µn+2n . n k

k=0

(See Chu and de Donno [316, Theorem 1]) 10. Show that ∞ X k=1


22k−1 − 1 ζ (2k) (2p)2k (m + k)    p 2m 1 − γ (p) + (−1)m (2m)! 1 − 2−2m−1 ζ (2m + 1) 4m π     p−1 m (2m)! X (−1)k (2π )−2k X (2` + 1)π 2` + 1 − cos ζ 2k + 1, 2p (2m − 2k)! p 2p

=−

k=1

`=0

    p−1 m X (−1)k (2π )−2k X (2` + 1)π 2` + 1 π sin ζ 2k, − (2m)! p (2m − 2k + 1)! p 2p k=1

`=0

(m ∈ N; p ∈ N \ {1}) and ∞ X k=1


22k−1 − 1 1 1 ζ (2k) = − − γ (p) 4(2m + 1) 2 (2p)2k (2m + 2k + 1)

+

    p−1 m (2m + 1)! X (−1)k (2π )−2k X (2` + 1)π 2` + 1 sin ζ 2k + 2, 8pπ (2m − 2k)! p 2p k=0

`=0

    p−1 m (2m + 1)! X (−1)k (2π )−2k X (2` + 1)π 2` + 1 − cos ζ 2k + 1, 4p (2m − 2k + 1)! p 2p k=1

`=0

(m ∈ N0 ; p ∈ N \ {1}),

396

Zeta and q-Zeta Functions and Associated Series and Integrals

where γ (p) is given by γ (p) =

p−1 ∞ X X

  (2` + 1)π 1 cos . 2pj + 2` + 1 p

j=0 `=0

(Cho et al. [257, Equations (40) and (41)]) 11. Show that n X

1 − 5 j Hj + 5 j Hn−j

j=0

   n  2  X
n 5 n n+j = (−1)n , j j j j=0

where Hn is the harmonic number. (Paule and Schneider [890, Eq. (5)]) 12. For n ∈ N, define F(n) =

   n  X n+j 2 n 2 j=1

j

j


1 + 2 j Hn+j + 2 j Hn−j − 4 j Hj .

Show that F(n) = 0. (Ahlgren and Ono [14, Theorem 7]) 13. A regularized version of Riemann Zeta function is ζ (s) −

  ∞ X s 1 (−1)n bn . = n s−1 n=0

Show that bn = n (1 − γ − Hn−1 ) −

∞   1 X n + (−1)k ζ (k). 2 k k=2

(Flajolet and Vepstas [455, Eq. (2)]) 14. Show that Zt

log(sin θ ) dθ = t(log t − 1) −

∞ X (−1)k+1 B2k (2t)2k+1 4k(2k + 1)!

(|t| < π).

k=1

0

(Monegato and Strozzi [839, Eq. (17)]) 15. Show that  3   n k   X X n   = n 8n + 8n − 3n 2n 2n . j 2 4 n k=0

j=0

(Zhang and Wang [1256, Eq. (1)]) 16. Let {n }n∈N0 be a suitably bounded sequence of complex numbers. Also let the parameters λ and µ be constrained by <(λ − µ) > 0

(λ ∈ / Z− 0 ).

Series Involving Zeta Functions

397

Then show that ∞ ∞ ∞ X (ν)n X k zk X  k zk = [ψ(λ + k) − ψ(λ − ν + k)] n(λ)n (λ + n)k k! (λ)k k! n=1

k=0

k=0

and ∞ ∞ ∞ X zk (ν)n X (ν + n)k k zk X (ν)k k = [ψ(λ + k) − ψ(λ − ν)] , n(λ)n (λ + n)k k! (λ)k k! n=1

k=0

k=0

provided that the series involved converge absolutely. (Nishimoto and Srivastava [863, p. 104]; see also 3.5(12) and R. Srivastava [1115] for further results of these classes) 17. Derive the following hypergeometric identity: 1 + a, 2 + 12 a, 1 + b, 1 + c, 2 + 2a − b − c + N, 1 − N, 1, 1;

 8 F7 

2 + a, 1 +

=

1 2 a, 2 + a − b, 2 + a − c, 1 + b + c − a − N, 2 + a + N, 2;

 1

(1 + a)(1 + a − b)(1 + a − c)(b + c − a − N)(1 + a + N) (2 + a)bc(1 + 2a − b − c + N)N  · ψ(1 + a) − ψ(1 + a + N) + ψ(1 + a − b + N) − ψ(1 + a − b) + ψ(1 + a − c + N) − ψ(1 + a − c) + ψ(1 + a − b − c) − ψ(1 + a − b − c + N)



(N ∈ N),

provided that no zeros appear in the denominator of either member. (Srivastava [1079, p. 80, Eq. (1.3)]) 18. Prove the following identity: n X k=1

k

n
m
k k n+k
m+k
k k



 1 3 1 1 2 Hk−1 + Hk2 = e + e0 e1 + e2 , 12 0 4 6

where m, n ∈ N; n 5 m and (`+1)

(`+1) e` := Hn(`+1) + Hm − Hn+m

(` ∈ N0 ). (Choi [266, Eq. (3.20)])

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4 Evaluations and Series Representations

Based essentially on some of the most recent works on the subject by Srivastava [1084, 1086], Srivastava and Tsumura [1111] and others, this chapter aims at investigating rather systematically several interesting evaluations and representations of the Riemann Zeta function ζ (s) when s ∈ N \ {1}. We begin by presenting some of the methods of evaluation of ζ (2n) (n ∈ N). We then proceed to develop various families of rapidly convergent series representations for ζ (2n + 1) (n ∈ N). Finally, in one of the many computationally useful special cases considered here, we observe that ζ (3) can be represented by means of a series that converges much more rapidly than that in Euler’s celebrated formula 3.1(8), as well as the series 4.2(2) used by Ape´ ry [56] in his proof of the irrationality of ζ (3). Symbolic and numeric computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of this series are capable of producing an accuracy of seven decimal places.

4.1 Evaluation of ζ (2n) The solution of the so-called Basler problem (cf., e.g., Spiess [1061, p. 66]): ζ (2) =

∞ X 1 π2 = 6 k2

(1)

k=1

was first found in 1736 by Leonhard Euler (1707–1783), although Jakob Bernoulli (1654–1705) and Johann Bernoulli (1667–1748) did their utmost to sum the series in (1). In fact, the former of these Bernoulli brothers did not live to see the solution of the problem, and the solution became known to the latter soon after Euler found it (see, for details, Knopp [676, p. 238]). The Basler problem (1) has been solved in the mathematical literature in many different ways. In addition to numerous papers containing elementary proofs of (1), many of which are referenced by Stark [1125], there are a fairly large number of books on complex analysis and advanced calculus in which (1) is proven by using Cauchy’s residue calculus, Weierstrass’s product theorem, Parseval’s theorem, Fourier series expansions and so on. Some of these books were referred to by Choe [258], who seems to have rediscovered one of Euler’s remarkably elementary proofs of (1) Zeta and q-Zeta Functions and Associated Series and Integrals. DOI: 10.1016/B978-0-12-385218-2.00004-9 c 2012 Elsevier Inc. All rights reserved.

400

Zeta and q-Zeta Functions and Associated Series and Integrals

(cf., e.g., Ayoub [81, p. 1079]), which is based on the integration of the following Taylor series expansion of arcsin x near x = 0: ∞ X 1 · 3 · 5 · · · (2k − 1) x2k+1 arcsin x = x + 2 · 4 · 6 · · · (2k) 2k + 1

(−1 5 x 5 1)

(2)

k=1

with respect to t (where x = sin t), by means of Wallis’ integral formula in the form: Zπ/2 sin2k+1 t dt = 0

2 · 4 · 6 · · · (2k) 1 · 3 · 5 · · · (2k + 1)

(k ∈ N).

(3)

Motivated largely by the aforementioned (Euler’s) proof of (1) detailed by Choe [258], Choi and Rathie [279] gave an essentially analogous derivation of (1), by appealing to the Gauss summation theorem 1.4(7). Subsequently, Choi, Rathie and Srivastava [281] showed how the derivation of (1) by Choi and Rathie [279] can be accomplished without using the Gauss summation theorem 1.4(7). They also presented several other evaluations of ζ (2) and related sums, including (for example) a fairly straightforward evaluation based on the theory of hypergeometric series.

Elementary and Hypergeometric Evaluations of ζ (2) Starting from the elementary integral: 1 (arcsin x)2 = 2

Zx 0

arcsin t dt √ 1 − t2

(4)

and replacing arcsin t by its Taylor series expansion given by (2), it is not difficult to obtain the representation: ∞   −2k Z x X 1 2k 2 t2k+1 2 (arcsin x) = dt √ 2 k 2k + 1 1 − t2 k=0

(−1 5 x 5 1)

(5)

0

or, equivalently, 1 (arcsin x)2 = 2

∞ X k=0

 

1 2 k

k!

1 2k + 1

Zx 0

t2k+1 dt √ 1 − t2

(−1 5 x 5 1).

(6)

Choi and Rathie [279, p. 396] express the second member of (6) as a hypergeometric
2 −1/2 to evaluate the integral, 2 F1 series, by using the binomial expansion of 1 − t set x = 1 and then apply the Gauss summation theorem 1.5(7).

Evaluations and Series Representations

401

The use of the Gauss summation theorem 1.5(7) in the above derivation of (1) by Choi and Rathie [279] √ can easily be avoided. Indeed, in the special case of (6) when x = 1, if we set t = u and apply the known integral formula 1.1(39) and 1.1(42), we readily find from (6) that ∞ X k=0

π2 1 = , 8 (2k + 1)2

(7)

which, in view of 2.3(1), is equivalent to (1). By virtue of the relationships 2.5(26) and    
1 1 8 1, s, = ζ s, = 2s − 1 ζ (s), 2 2

(8)

2.5(17) immediately yields the hypergeometric representations: ζ (n, a) = a−n n+1 Fn (1, a, . . . , a; a + 1, . . . , a + 1; 1)

(n ∈ N \ {1})

(9)

(n ∈ N \ {1}).

(10)

and ζ (n) =



2n n 2 −1



  1 1 3 3 F 1, , . . . , ; , . . . , ; 1 n+1 n 2 2 2 2

In particular, by setting n = 2 in (10), we have   4 1 1 ζ (2) = 3 F2 1, , ; 23 , 32 ; 1 , 3 2 2

(11)

which can be evaluated by applying Dixon’s summation theorem 3.5(4) with a=1

and b = c =

1 2

or by applying Whipple’s summation theorem 3.5(6) with 1 a=b= , 2

c=1

3 and d = e = . 2

We, thus, find from (11) and either one of the above applications that   1 π2 4 n  3 o3 0 2 0 = . ζ (2) = 3 2 6

(12)

Next, by appealing directly to the definition 2.3(1) or (9) with a = 1, we obtain the hypergeometric representation: ζ (n) = n+1 Fn (1, . . . , 1; 2, . . . , 2; 1)

(n ∈ N \ {1}) .

(13)

402

Zeta and q-Zeta Functions and Associated Series and Integrals

The representations (10) and (13), together, yield the hypergeometric transformation:   1 3 1 3 , . . . , ; 1 F , . . . , ; 1, n+1 n 2 2 2 2 (14) −n = (1 − 2 ) n+1 Fn (1, . . . , 1; 2, . . . , 2; 1) (n ∈ N \ {1}) , which, for n = 2, assumes the form:   1 1 3 3 3 3 F2 1, , ; 2 , 2 ; 1 = 3 F2 (1, 1, 1; 2, 2; 1). 2 2 4

(15)

We remark in passing that, by making use of contiguous analogues of Dixon’s summation theorem 3.5(4) and Whipple’s summation theorem 3.5(6), which were given recently by Lavoie et al. [731, p. 268, Eq. (2); 732, p. 294, Eq. (4)] we can easily express the sums of many series [analogous to those occurring in (1) and (7)] in terms of ζ (2). The details are fairly straightforward. Since ζ (n) =

Z1

Z1 ···

0

0

dt1 . . . dtn 1 − t1 . . . tn

(n ∈ N \ {1}),

(16)

it is not difficult to provide yet another solution of the Basler problem (1), by using the following change of variables: v+u t1 = √ 2

v−u and t2 = √ 2

in the double integral resulting from (16) when n = 2 (cf., e.g., Ojha and Singh [874]).

The General Case of ζ (2n) Following Srivastava [1086], let us begin, by recalling the following summation formula for the Bernoulli polynomials (cf. Hansen [531, p. 342, Entry (50.11.10)]): [n/2] X k=0

(−n)2k 1 B2k (x)Bn−2k (y) = (x + y − 1)nBn−1 (x + y) (2k)! 2 + (−1)n (x − y)nBn−1 (x − y)   − (n − 1) Bn (x + y) + (−1)n Bn (x − y) ,

(17)

which, upon setting x = 1 and y = 0 (and replacing n by 2n), yields n   X 2n k=0

2k

B2k B2n−2k = −nB2n−1 − (2n − 1)B2n

(n ∈ N0 ) .

(18)

Evaluations and Series Representations

403

Finally, by transposing the terms for k = 0 and k = n in (18) to the right-hand side and making use of 1.7(7), we obtain n−1   X 2n

2k

k=1

B2k B2n−2k = −(2n + 1)B2n

(n ∈ N \ {1}),

(19)

which is, in view of the relationship 2.3(18), equivalent to 2.3(20), which (as noted there) can also be used to evaluate ζ (2n) (n ∈ N \ {1}) with (1). To give a direct proof of the summation formula 2.3(20), let us put (cf. Prudnikov [917]) (x) :=

∞ X

  exp −π 2 k2 x

(x > 0),

(20)

k=1

so that Zx 0


∞ 3 X exp −π 2 k2 x , (x − t)(t)dt = x(x) + 2 π 2 k2

(21)

k=1

which, by virtue of (1), readily yields Zx

(x − t)(t)dt =

lim

x→0

1 4

(22)

0

and Zx

(x − t)(t)dt = O xe−x




(x → ∞).

(23)

0

Making use of (21) and (23), it is easily seen that Z∞Z∞ Is := (x + y)s−2 (x)(y)dx dy 0 0

Z∞ Zx s−2 = x dx (x − t)(t)dt 0

0

  0(s − 1) = s + 12 ζ (2s) π 2s

(<(s) > 1).

(24)

404

Zeta and q-Zeta Functions and Associated Series and Integrals

Conversely, since Z∞ 0(s) ts−1 (t)dt = 2s ζ (2s) π

  <(s) > 0; s 6= 21 ,

(25)

0

which is an immediate consequence of the definition (20) and 2.3(1), an appeal to the Binomial theorem for (x + y)n−2 in the form: (x + y)n−2 =

 n−1  X n − 2 k−1 n−k−1 x y k−1

(n ∈ N \ {1})

(26)

k=1

would lead us also to n−1

(n − 2)! X ζ (2k)ζ (2n − 2k) In = π 2n

(n ∈ N \ {1}).

(27)

k=1

Now, the summation formula 2.3(20) follows upon equating the two values of In given by (24) (with s = n) and (27). Remark 1 The recursion formula (19) is a well-known (rather classical) result for Bernoulli numbers. It appears (for example) in Nielsen’s book [861] and was also derived independently by Underwood [1177]. The equivalent result 2.3(20) was proven, by elementary methods by Williams [1229, p. 20, Theorem I] (see also Apostol [63, p. 427]). Remark 2 In terms of the L-function defined by [cf. Eq. 2.5(1)] L(s) =

∞   X (−1)k −s 1 = 2 8 −1, s, 2 (2k + 1)s

(<(s) > 0),

(28)

k=0

Williams [1229, p. 22, Theorem II] gave an interesting companion of the result 2.3(20) in the form: n X

   L(2k − 1)L(2n − 2k + 1) = n − 12 1 − 2−2n ζ (2n),

(29)

k=1

which appears erroneously in Hansen [531, p. 357, Entry (54.7.1)]. Since L(1) is the π well-known Gregory series for (with L(2) being the familiar Catalan constant G), 4 by setting n = 1 in (29), we immediately obtain ζ (2) =

8 π2 {L(1)}2 = . 3 6

(30)

Evaluations and Series Representations

405

  Remark 3 In view of the constraint <(s) > 0 s 6= 12 associated with the Mellin transform in (25), an earlier attempt by Kalla and Villalobos [623] to extend the summation formula 2.3(20), by expressing Is as a non-terminating (infinite) sum analogous to (27), cannot be justified. Thus, the main result of Kalla and Villalobos [623, p. 17, Eq. (17)] holds true only in the finite form given already by the well-known result 2.3(20).

4.2 Rapidly Convergent Series for ζ (2n + 1) On the subject of series representations for ζ (3), in addition to Euler’s result 3.1(8), Chen and Srivastava [252] gave many series expressions for ζ (3), which are more rapidly convergent than that in 3.1(8), including ζ (3) = −

∞ ζ (2k) 8π 2 X . 5 (2k + 1)(2k + 2)(2k + 3) 22k

(1)

k=0

And, as pointed out by (for example) Chen and Srivastava [252, pp. 180–181], another interesting series representation: ∞

ζ (3) =

5 X (−1)k−1  , 2k 2 k=1 k3 k

(2)

which played a key roˆ le in Ape´ ry’s proof of the irrationality of ζ (3) (see Ape´ ry [56]), was proven independently by (among others) Hjortnaes [559], Gosper [496] and Ape´ ry [56]. Furthermore, in their recent work on the Ray-Singer torsion and topological field theories, Nash and O’Connor [855, 856] obtained a number of remarkable integral expressions for ζ (3), including (for example) the following result [856, p. 1489 et seq.]: 8 2π 2 log 2 − ζ (3) = 7 7

Zπ/2 z2 cot z dz.

(3)

0

Since (Erde´ lyi et al. [421, p. 51, Eq. 1.20(3)]) z cot z = −2

∞ X k=0

ζ (2k)

 z 2k π

(|z| < π ),

(4)

the result (3) is obviously equivalent to the series representation (cf. Da¸browski [362, p. 202]; see also Chen and Srivastava [252, p. 191, Eq. (3.19)]): ! ∞ X 2π 2 ζ (2k) ζ (3) = log 2 + . (5) 7 (k + 1)22k k=0

406

Zeta and q-Zeta Functions and Associated Series and Integrals

Moreover, by integrating by parts, it is easily seen that Zπ/2 Zπ/2 2 z cot z dz = −2 z log sin z dz, 0

(6)

0

so that the result (3) is equivalent also to the integral representation: 16 2π 2 log 2 + ζ (3) = 7 7

Zπ/2 z log sin z dz,

(7)

0

which was proven in the aforementioned 1772 paper by Euler (cf., e.g., Ayoub [81, p. 1084]; see also Srivastava [1086]). If follows from 3.2(7) that (cf., e.g., Hansen [531, p. 359]; see also Srivastava [1084] where other references are also cited) ∞ X (s)2k 1 ζ (s + 2k, a) t2k = [ζ (s, a − t) + ζ (s, a + t)] (2k)! 2

(|t| < |a|) .

(8)

k=0

In view of 2.3(3), the special case of the identity (8) when a = 1 and t = rewritten in the form:

1 m

can be

∞ X (s)2k ζ (s + 2k) (2k)! m2k k=0

 s (2 − 1) ζ (s) − 2s−1       m−2 = 1 X  j  s  s   ζ s,   2 (m − 1)ζ (s) − m − m

(m = 2)

(9)

(m ∈ N \ {1, 2}),

j=2

where (as usual) an empty sum is interpreted as nil. In addition to the case m = 2, the formula (9) simplifies also in the cases when m = 3, 4 and 6, and we, thus, obtain the identities: ∞ X  (s)2k ζ (s + 2k) 1  s = (3 − 1) ζ (s) − 3s , (2k)! 2 32k

(10)

∞ X  (s)2k ζ (s + 2k) 1  s = (4 − 2s ) ζ (s) − 4s 2k (2k)! 2 4

(11)

k=0

k=0

Evaluations and Series Representations

407

and ∞ X  (s)2k ζ (s + 2k) 1  s = (6 − 3s − 2s + 1) ζ (s) − 6s , 2k (2k)! 2 6

(12)

k=0

respectively. Identities of this kind seem to have first appeared in the work of Ramaswami [966], who actually proved the cases m = 2, 3 and 6 of the general result in (9). Each of these three identities of Ramaswami [966] can also be found in the work of Hansen [531, p. 357], who referred to Apostol [61] as his source for the identities (10) and (12) only. As a matter of fact, Apostol [61] reproduced the identities (10) and (12) from Ramaswami’s work [966] and then proved an interesting arithmetical generalization of these identities (see also Klusch [673, p. 520]). In its slightly variant form: ∞ X (s + 1)2k ζ (s + 2k) = (2s − 2) ζ (s), (2k)! 22k

(13)

k=1

the case m = 2 of the general result (9) was applied by Zhang and Williams [1251] (and, subsequently, by Cvijovic´ and Klinowski [351]) with a view to finding two seemingly different series representations for ζ (2n + 1) (n ∈ N). Srivastava [1084] obtained much more rapidly converging series representations for ζ (2n + 1) (n ∈ N), chiefly by appealing appropriately to each of the aforementioned cases (m = 2, 3, 4 and 6) of the general result (9). Following his work (Srivastava [1084]), we begin, here, with the case m = 2 of the general result (9). Upon separating the first n + 1 terms of the series occurring on the left-hand side, if we transpose the terms for k = 0 and k = n to the right-hand side, we obtain n−1 ∞ X X (s)2k ζ (s + 2k) (s)2k ζ (s + 2k) + 2k (2k)! (2k)! 2 22k k=1

k=n+1

(14)

(s)2n ζ (s + 2n) = (2s − 2)ζ (s) − 2s−1 − , (2n)! 22n which readily yields the identity: n−1 ∞ X X (s)2k 2(n−k) (s)2n+2k ζ (s + 2n + 2k) 2 ζ (s + 2k) + (2k)! (2n + 2k)! 22k k=1

k=1

(s)2n = 22n (2s − 2) ζ (s) − 2s+2n−1 − ζ (s + 2n) (2n)!

(n ∈ N).

(15)

408

Zeta and q-Zeta Functions and Associated Series and Integrals

Now, we apply the functional equation 2.3(12) in the first term on the right-hand side of (15) and divide both sides by s + 2n. We, thus, find that   X ∞ n−1 X (s)2n (s + 2n +1)2k−1 ζ (s + 2n + 2k) (s)2k 2(n−k) ζ (s + 2k) 2 + (2k)! s + 2n (2n + 2k)! 22k k=1 k=1     sin 12 πs  (16) = 2s+2n (2s − 2) π s−1 0(1 − s) ζ (1 − s)  s + 2n    s+2n−1 2 + (s)2n (2n)! ζ (s + 2n) (s 6= −2n; n ∈ N). − s + 2n It is easy to see from 1.6(10) that d {(s)2n } = −(2n)! H2n ds s=−2n

(n ∈ N),

(17)

where Hn denotes the familiar harmonic numbers, defined by 3.2(36). We observe also that the following limit formula:  lim

s→−2n

 ζ (s + 2k) (−1)n−k = (2n − 2k)! ζ (2n − 2k + 1) s + 2n 2(2π)2(n−k) (k = 1, . . . , n − 1; n ∈ N \ {1})

(18)

is needed in the first sum on the left-hand side of (16) only when this sum is nonzero (that is, only when n ∈ N \ {1}). Furthermore, by l’Hoˆ pital’s rule, we have    s+2n−1 + (s)2n ζ (s + 2n)    2   (2n)! lim  s→−2n  s + 2n       (19) ζ (s + 2n) (s)2n 0 d s+2n−1 + ζ (s + 2n) = 2 log 2 + {(s)2n } · ds (2n)! (2n)! s=−2n 1 = (H2n − log π) (n ∈ N). 2 Finally, by letting s → −2n in (16) and making use of the limit relationships (18) and (19), we obtain Srivastava’s first series representation for ζ (2n + 1): ζ (2n + 1) = (−1)

n−1

" n−1 (2π)2n H2n − log π X (−1)k ζ (2k + 1) + (2n)! (2n − 2k)! 22n+1 − 1 π 2k k=1 (20) # ∞ X (2k − 1)! ζ (2k) +2 (n ∈ N). (2n + 2k)! 22k k=1

Evaluations and Series Representations

409

In precisely the same manner, we can apply the identities (10), (11) and (12) to prove the following additional series representations for ζ (2n + 1) (Srivastava [1084, p. 389]):    2 n−1 2n H − log X 2n 3π (−1)k ζ (2k + 1) 2(2π)  + ζ (2n + 1) = (−1)n−1 2n+1    (2n)! (2n − 2k)! 2 2k 3 −1 k=1 3π  ∞ X (2k − 1)! ζ (2k)  +2  (n ∈ N); (2n + 2k)! 32k k=1

(21) 

ζ (2n + 1) = (−1)

n−1

+

n−1 X k=1



H − log 12 π 2(2π)2n  2n  (2n)! 24n+1 + 22n − 1



 ∞ X ζ (2k + 1) (2k − 1)! ζ (2k)     +2 (2n − 2k)! 1 2k (2n + 2k)! 42k k=1 π 2 (−1)k

(n ∈ N); (22)



ζ (2n + 1) = (−1)

n−1

+

n−1 X k=1



H − log 13 π 2(2π)2n  2n  (2n)! 32n (22n + 1) + 22n − 1 ∞ X





(2k − 1)! ζ (2k)  (−1)k ζ (2k + 1)   2k + 2 (2n − 2k)! 1 (2n + 2k)! 62k k=1 π 3

(n ∈ N). (23)

Remarks and Observations The series representation (20) is markedly different from each of the series representations for ζ (2n + 1), which were given earlier by Zhang and Williams [1251, p. 1590, Eq. (3.13)] and (more recently) by Cvijovic´ and Klinowski [351, p. 1265, Theorem A]. Since ζ (2k) → 1 as k → ∞, the general term in the series representation (20) has the order estimate:   O 2−2k · k−2n−1 (k → ∞; n ∈ N), whereas the general term in each of these earlier series representations has the order estimate:   O 2−2k · k−2n (k → ∞; n ∈ N).

410

Zeta and q-Zeta Functions and Associated Series and Integrals

By suitably combining (20) and (22), it is fairly straightforward to obtain the series representation:

ζ (2n + 1) = (−1)

n−1

+

" log 2 2(2π)2n (22n − 1)(22n+1 − 1) (2n)!

n−1 X (−1)k (22k − 1) ζ (2k + 1) (2n − 2k)! π 2k k=1 ∞ X

−2

k=1

(24)

(2k − 1)! (22k − 1) ζ (2k) (2n + 2k)! 24k

# (n ∈ N).

Making use of the relationship given in Problem 11 of Chapter 2, the series representation (24) can immediately be put in the form:

ζ (2n + 1) = (−1)

n−1

+

" 2(2π)2n log 2 2n 2n+1 (2 − 1)(2 − 1) (2n)!

n−1 X (−1)k (22k − 1) ζ (2k + 1) (2n − 2k)! π 2k

(25)

k=1

# ∞ 1 X (−1)k−1  π 2k + E2k−1 (0) 2 (2n + 2k)! 2

(n ∈ N),

k=1

which is a slightly modified (and corrected) version of a result proven in a significantly different way by Tsumura [1169, p. 383, Theorem B]. Another interesting combination of the series representations (20) and (22) leads to the following variant of Tsumura’s result (24) or (25):

ζ (2n + 1) = (−1)n−1

+

π 2n



22n+1 − 1



H2n − log



1 4π



(2n)!

n−1 X (−1)k (22k+1 − 1) ζ (2k + 1) (2n − 2k)! π 2k k=1 ∞ X

−4

k=1

(2k − 1)! (22k−1 − 1) ζ (2k) (2n + 2k)! 24k

(26) # (n ∈ N),

which is essentially the same as the determinate expression for ζ (2n + 1), derived recently by Ewell [438, p. 1010, Corollary 3] by employing an entirely different technique from Srivastava’s [1084].

Evaluations and Series Representations

411

Other similar combinations of the series representations (20) to (23) would yield the following companions of Ewell’s result (26):

ζ (2n + 1) = (−1)n−1

+

2(2π)2n



(22n+1 − 1)(32n + 1)



H2n − log



1 6π



(2n)!

n−1 X (−1)k (22k+1 − 1) ζ (2k + 1)  2k (2n − 2k)! 2 k=1 3π

(27)

 ∞ X (2k − 1)!(22k−1 − 1) ζ (2k)  −4 (2n + 2k)! 62k

(n ∈ N);

k=1

ζ (2n + 1) = (−1)n−1

+

2(2π)2n



2H2n − log

(22n + 1)(32n+1 − 1)



(2n)!



π2 27



n−1 X (−1)k (32k+1 − 1) ζ (2k + 1) (2n − 2k)! π 2k

(28)

k=1

 ∞ 2k−1 X (2k − 1)! (3 − 1) ζ (2k)  −6 (2n + 2k)! 62k

(n ∈ N);

k=1

ζ (2n + 1) = (−1)n−1

+



2(2π)2n



H2n − log

32n+2 − 22n+3 + 1



(2n)!

8π 27



n−1 X (−1)k (32k+1 − 22k+1 ) ζ (2k + 1) (2n − 2k)! (2π)2k

(29)

k=1

 ∞ X (2k − 1)! (32k−1 − 22k−1 ) ζ (2k)  − 12 (2n + 2k)! 62k

(n ∈ N);

k=1

   2n H2n − log 27π 128 2(2π)  ζ (2n + 1) = (−1)n−1 4n+3 (2n)! 2 + 22n+2 − 32n+2 − 1 n−1 X (−1)k (42k+1 − 32k+1 ) ζ (2k + 1) + (2n − 2k)! (2π)2k

(30)

k=1

 ∞ 2k−1 2k−1 X (2k − 1)! (4 −3 ) ζ (2k)  − 24 (2n + 2k)! 122k k=1

(n ∈ N),

412

Zeta and q-Zeta Functions and Associated Series and Integrals

and

ζ (2n + 1) = (−1)n−1

+



2(2π)2n



H2n − log

32n+1 (22n + 1) − 24n+2 + 22n − 1



(2n)!

n−1 X (−1)k (32k+1 − 22k+1 ) ζ (2k + 1) (2n − 2k)! π 2k

4π 27



(31)

k=1

 ∞ X (2k − 1)! (32k−1 − 22k−1 ) ζ (2k)  − 12 (2n + 2k)! 122k

(n ∈ N).

k=1

Next, we recall another identity similar to (8) (see Srivastava [1084, p. 386, Eq. (1.7)]): ∞ X 1 (s)2k+1 ζ (s + 2k + 1, a) t2k+1 = [ζ (s, a − t) − ζ (s, a + t)] (2k + 1)! 2

(|t| < |a|) .

k=0

(32) By setting t = 1/m and differentiating both sides with respect to s, we find from (32) that ∞ X k=0

  2k X 1  (s)2k+1  0 ζ (s + 2k + 1, a) + ζ (s + 2k + 1, a) s+j (2k + 1)! m2k j=0

     m d 1 1 = ζ s, a − − ζ s, a + 2 ds m m

(33)

(m ∈ N \ {1}).

In particular, when m = 2, (33) immediately yields ∞ X k=0

  2k X (s)2k+1 1 ζ 0 (s + 2k + 1, a) + ζ (s + 2k + 1, a)  s+j (2k + 1)! 22k j=0

(34)

    1 1 −s = − a− log a − . 2 2 By letting s → −2n − 1 (n ∈ N) in the further special of this last identity (34) when a = 1, Wilton [1233, p. 92] obtained the following series representation for ζ (2n + 1)

Evaluations and Series Representations

413

(see also Hansen [531, p. 357, Entry (54.6.9)]): " n−1 X (−1)k ζ (2k+1) n−1 2n H2n+1 − log π ζ (2n + 1) = (−1) π + (2n + 1)! (2n−2k+1)! π 2k k=1 # ∞ X (2k − 1)! ζ (2k) +2 (n ∈ N), (2n + 2k + 1)! 22k

(35)

k=1

which may be compared with the series representation (20). As a matter of fact, since (2k)! (2k − 1)! (2k − 1)! = − 2n (2n + 2k)! (2n + 2k − 1)! (2n + 2k)!

(n, k ∈ N),

it is not difficult to deduce from (20) and (35) (with n replaced by n − 1) that " n−1 X (−1)k−1 k ζ (2k + 1) (2π)2n n ζ (2n + 1) = (−1) (2n − 2k)! n(22n+1 − 1) π 2k k=1 # ∞ X (2k)! ζ (2k) + (n ∈ N), (2n + 2k)! 22k

(36)

(37)

k=0

which is precisely the aforementioned main result of Cvijovic´ and Klinowski [351, p. 1265, Theorem A]. As a matter of fact, in view of the derivative formula 2.3(22), the series representation (37) is essentially the same as a result given earlier by Zhang and Williams [1251, p. 1590, Eq. (3.13)] (see also Zhang and Williams [1251, p. 1591, Eq. (3.16)]), where an obviously more complicated (asymptotic) version of (37) was proven, by applying the same identity (13) above. Observing also that (2k)! (2k − 1)! (2k − 1)! = − (2n + 1) (2n + 2k + 1)! (2n + 2k)! (2n + 2k + 1)!

(n, k ∈ N),

(38)

we obtain yet another series representation for ζ (2n + 1), by applying (20) and (35): " n−1 X (−1)k−1 k ζ (2k + 1) 2(2π)2n n ζ (2n + 1) = (−1) 2n (2n − 2k + 1)! r(2n − 1) 2 + 1 π 2k k=1 # ∞ X (2k)! ζ (2k) + (n ∈ N), (2n + 2k + 1)! 22k k=0

(39) which provides a significantly simpler (and much more rapidly convergent) version of the other main result of Cvijovic´ and Klinowski [351, p. 1265, Theorem B]: ζ (2n + 1) = (−1)n

∞ 2(2π)2n X ζ (2k) n,k 2k (2n)! 2 k=0

(n ∈ N),

(40)

414

Zeta and q-Zeta Functions and Associated Series and Integrals

where the coefficients n,k are given explicitly by n,k :=

2n   X 2n

j

j=0

B2n−j (j + 2k + 1)(j + 1) 2j

(n ∈ N; k ∈ N0 ),

(41)

in terms of the Bernoulli numbers (see Section 1.6). The definition (41) can be rewritten at once in the form: n   X 2n

n,k =

2j

j=0

B2n−2j 1 − (2j + 2k + 1)(2j + 1) 22j (2n + 2k) 22n

(n ∈ N; k ∈ N0 ), (42)

or, equivalently, n,k = (−1)

n−1

n 2(2n)! X (−π 2 ) j ζ (2n − 2j) (2j + 1)!(2j + 2k + 1) (2π)2n j=0

1 − (2n + 2k) 22n

(43)

(n ∈ N; k ∈ N0 ),

by virtue of the relationship 2.3(18). Combining the partial fractions occurring in (42) or (43), it is easily seen that n   Q (2k + 2` + 1)−1 n   X 2n 2n + 2k 2n−2j `=0  n,k = 2 B2n−2j  2n 2j 2j + 1 (2n + 2k) 2 j=0

(44)

 ·

n Y

(2k + 2` + 1) −

`=0 (`6=j)

n Y `=0

 (2k + 2` + 1) 

(n ∈ N; k ∈ N0 ).

In view of the identity: n   2n−2j X B2n−2j 2n 2 j=0

2j

2j + 1

=1=

n   X 2n j=0

22j B2j , 2j 2n − 2j + 1

(45)

which is due essentially to Euler (cf., e.g., Riordan [978, p. 123, Problem 12]), the expression inside brackets in (44) is a polynomial in k of degree n (not n + 1), and, therefore, n,k = O(k−2 )

(k → ∞; n ∈ N).

(46)

Evaluations and Series Representations

415

It follows from (46) that the general term in (40) has the order estimate: O(2−2k · k−2 )

(k → ∞),

(47)

whereas the general term in our series representation (39) has precisely the same order estimate: O(2−2k · k−2n−1 )

(k → ∞; n ∈ N)

(48)

as that in (20). Thus, even in the special case when n = 1, the series representing ζ (3) converges faster in (39) than in (40). Various known series representations for ζ (2n + 1) (n ∈ N) of other types include those given (for example) by Ramanujan [965] (see also Berndt [121]), Glaisher [489] (see also Hansen [531, p. 359]), Koshliakov [695], Leshchiner [747], Grosswald [514, 515], Terras [1147], Cohen [334], Butzer et al. [197, 198], Da¸browski [362] and others (see, e.g., Berndt [122, pp. 275 and 276]). We conclude this section by remarking that a particular case of the series representation (23) when n = 1 was proven, by an entirely different method by Zhang and Williams [1251, p. 707, Theorem 9]. Furthermore, the following particular case of (37) when n = 1 (see 3.1(8)): ζ (3) = −

∞ ζ (2k) 4π 2 X , 7 (2k + 1)(2k + 2) 22k

(49)

k=0

which is contained in a 1772 paper entitled Exercitationes Analyticae by Euler (see, e.g., Ayoub [81, pp. 1084–1085]), was rediscovered by Ramaswami [966] and (more recently) by Ewell [436]. In fact, Euler’s formula (49) was reproduced by Srivastava [1072, p. 7, Eq. (2.23)] from the work of Ramaswami [966]. In the current mathematical literature, however, Euler’s formula (49) is being attributed to Ewell [436].

4.3 Further Series Representations There are many further known series representations for ζ (2n + 1) (n ∈ N), which converge much more rapidly than those given by the defining series in 2.3(1). For example, we have the series representation (see also Equation 4.2(35)): " ζ (2n + 1) = (−1)

n−1

π

2n

n−1

H2n+1 − log π X (−1)k ζ (2k + 1) + (2n + 1)! (2n − 2k + 1)! π 2k k=1 # ∞ X (2k − 1)! ζ (2k) +2 (n ∈ N), (2n + 2k + 1)! 22k k=1

(1)

416

Zeta and q-Zeta Functions and Associated Series and Integrals

which was given over seven decades ago by Wilton [1233, p. 92] (see also Hansen [531, p. 357, Entry (54.6.9)]), and the following result given recently by Srivastava [1083, p. 10, Eq. (42)] (see also Srivastava [1082, p. 5, Eq. (3.4)]):  π 2n



H2n+1 − log



1 2π



2(4n − 1) B2n+2 log 2 2 (2n + 1)! (2n + 2)!   1 22n+1 − 1 0 24n+3 0 − ζ (−2n − 1) − ζ −2n − 1, (2n + 1)! (2n + 1)! 4  n−1 ∞ k X X ζ (2k+1) (−1) (2k−1)! ζ (2k) +  (n ∈ N),   +2 (2n−2k+1)! 1 2k (2n+2k+1)! 42k k=1 k=1 π 2

ζ (2n +1) = (−1)n−1



+

(2) where (and in what follows) a prime denotes the derivative of ζ (s) or ζ (s, a) with respect to s, an empty sum is to be interpreted as nil and Hn denotes the familiar harmonic numbers defined by 3.2(36). Of the two seemingly analogous representations (1) and (2), the infinite series in (2) would obviously converge more rapidly with their general terms having the order estimates:   O k−2n−2 · m−2k (3) (k → ∞; n ∈ N; m = 2 and 4). Srivastava and Tsumura [1111] derived three (presumably new) members of the class of the series representations (1) and (2). The general terms of the infinite series occurring in these three members [given in (29), (30) and (31) below] have the order estimates:   O k−2n−2 · m−2k (4) (k → ∞; n ∈ N; m = 3, 4, 6), which exhibit the fact that each of the three series representations derived here for ζ (2n + 1) converges more rapidly than Wilton’s result (1) and two of them (cf. Equations (30) and (31) below) at least as rapidly as Srivastava’s result (2). Srivastava and Tsumura [1111] begin, by defining the sequence {γn (x)}∞ n=0 by means of the generating function (cf. Eq. 1.6(2)): ∞

F(x, t) :=

t − log x X tn = γ (x) n et − x n!


1 5 x 5 1 + c; c > 0 ,

(5)

n=0

so that, clearly, γn (1) = Bn

(n ∈ N0 ) .

(6)

Evaluations and Series Representations

417

Since the zeros of et − x are given by t = 2nπi + log x

(n ∈ Z) ,

(7)

the radius of convergence of the series in (5) is at least 2π. Hence, by the CauchyHadamard theorem for absolute convergence (cf., e.g., Whittaker and Watson [1225, p. 30]), we have Lemma 4.1 (Srivastava and Tsumura [2000a]) Let the sequence {γn (x)}∞ n=0 be defined by (5). Then there exists some nonnegative real number κ, such that  lim inf n→∞

|γn (x)| n!

−1/n

= 2π + κ

(κ = 0).

(8)

Following Srivastava and Tsumura [1111], we now consider the following Dirichlet series (cf. Eq. 2.3(1)):

ω(s, x) :=

∞ ∞ −n−1 X log x X x−n−1 x + ns s−1 ns−1


1 5 x 5 1 + c; c > 0 ,

(9)

n=1

n=1

so that, obviously, ω(s, 1) = ζ (s).

(10)

In case 1 < x 5 1 + c (c > 0), we can see that the function ω(s, x) is meromorphic, that is, holomorphic on the whole complex s-plane, except for a simple pole at s = 1 with residue log x . x−1 Lemma 4.2, below, provides an interesting generalization of the familiar relationship 2.3(10). Lemma 4.2 (Srivastava and Tsumura [1111]) Let γn (x) and ω(s, x) be defined by (5) and (9), respectively. Then ω(1 − n, x) = −

γn (x) n

(n ∈ N \ {1}) .

Making use of the generating function (5), we also have

(11)

418

Zeta and q-Zeta Functions and Associated Series and Integrals

Lemma 4.3 (Srivastava and Tsumura [1111]) For n ∈ N and |θ| < 2π (θ ∈ R),

(2n + 1)

∞ −k−1 ∞ −k−1 X X x x [sin(kθ) log x − θ cos(kθ)] sin(kθ) + 2n+2 k k2n+1 k=1

k=1

n−1

X (−1)k θ 2k+1 (2k + 1)! (−1)n θ 2n+1 log x = + (2n + 1)! x(x − 1) (2n − 2k) ω(2n − 2k + 1, x)

(12)

k=0

∞ X (−1)k θ 2k+1 γ2k−2n (x) (2k + 1)!

+

(1 < x 5 1 + c, c > 0).

k=n+1

By applying Lemma 4.3, Srivastava and Tsumura [1111] first proved that

(2n + 1)

∞ X sin(kθ) k=1

k2n+2

= 2(−1)n θ 2n+1

−θ

k=1 n−1 X k=1

−

∞ X k=0

∞ X cos(kθ)

k2n+1

− 2nθ ζ (2n + 1)

(−1)k · k ζ (2k + 1) (2n − 2k + 1)! θ 2k

(2k)! ζ (2k) (2n + 2k + 1)! (2π/θ)2k

(13)

! (|θ| < 2π ; θ ∈ R; n ∈ N).

Since each series in (13) is uniformly convergent with respect to θ on the open interval (−2π, 2π), by executing termwise differentiation in (13) with respect to θ , they showed that (Srivastava and Tsumura [1111, p. 328, Theorem 2])

2n

∞ ∞ X X cos(kθ) sin(kθ) + θ − 2n ζ (2n + 1) 2n+1 k k2n k=1

k=1

= 2(−1) θ

n 2n

n−1 ∞ X (−1)k · k ζ (2k + 1) X (2k)! ζ (2k) − 2k (2n − 2k)! (2n + 2k)! (2π/θ)2k θ k=1

!

(14)

k=0

(|θ | < 2π ; θ ∈ R; n ∈ N). Formula (14) was proven independently by Katsurada [638, p. 81, Theorem 1], by using the Melin transform techniques.

Evaluations and Series Representations

419

In their special cases when θ = π, Equations (13) and (14) readily yield a result given by Srivastava [1084, p. 393, Eq. (3.20)]: ζ (2n + 1) = (−1)n

X n−1 (−1)k−1 · k ζ (2k + 1) 2(2π)2n 2n (2n − 2k + 1)! (2n − 1) 2 + 1 π 2k k=1

(15) +

∞ X k=0

ζ (2k) (2k)! (2n + 2k + 1)! 22k



(n ∈ N)

and one of the two main results of Cvijovic´ and Klinowski [351, p. 1265, Theorem A]: X n−1 (2π)2n (−1)k−1 · k ζ (2k + 1) ζ (2n + 1) = (−1) (2n − 2k)! n(22n+1 − 1) π 2k n

k=1

(16) +

∞ X k=0

ζ (2k) (2k)! (2n + 2k)! 22k



(n ∈ N).

Remark 1 The series representation (15) was given by Srivastava [1082, p. 4, Eq. (2.5)] (see also Srivastava [1084, p. 393, Eq. (3.20)]. Indeed, as already observed by Srivastava [op. cit.], (15) provides a significantly simpler (and much more rapidly convergent) version of one of the two main results of Cvijovic´ and Klinowski [351, p. 1265, Theorem B]. Remark 2 The series representation (16) is the other main result of Cvijovic´ and Klinowski [351, p. 1265, Theorem A], whose companion was referred to in Remark 1 above. Remark 3 Since   1 (2k)! (2k − 1)! (2k − 1)! = − (2n + 2k + 1)! 2n + 1 (2n + 2k)! (2n + 2k + 1)!

(n, k ∈ N),

(17)

it is not difficult to obtain Wilton’s result (1), by combining the series representation (15) with another result 4.2(20) of Srivastava [1082, p. 1, Eq. (1.3)] (see also Srivastava [1084, p. 389, Eq. (2.9)]). With a view to applying (13) and (14) in their other special cases when 2 θ = π, 3

1 π 2

and

1 π, 3

(18)

420

Zeta and q-Zeta Functions and Associated Series and Integrals

we, now, recall several trigonometric sums given by Lemma 4 (Srivastava and Tsumura [1111]). For <(s) > 1,   ∞ cos 2 nπ X 3 31−s − 1 = ζ (s), (19) ns 2 n=1    −s   ∞ sin 2 nπ X √ 3 3 −1 1 −s = 3 ζ (s) + 3 ζ s, , (20) ns 2 3 n=1   ∞ cos 1 nπ X 2 = 2−s (21−s − 1) ζ (s), (21) ns n=1     ∞ sin 1 nπ X 2 1 −s 1−2s = (2 − 1) ζ (s) + 2 ζ s, , (22) ns 4 n=1   ∞ cos 1 nπ  X 3 1  1−s = 6 − 31−s − 21−s + 1 ζ (s), (23) s n 2 n=1

and ∞ sin X n=1



1 3 nπ ns

 =

      √ 3−s − 1 1 1 3 ζ (s) + 6−s ζ s, + ζ s, . 2 6 3

(24)

Srivastava and Tsumura [1111] first proved the following results:  2n   2n (3 − 1)π (−1)n π 1 n 2(2π) B2n + √ ζ 2n, ζ (2n + 1) = (−1) √ 3 n(32n+1 − 1) 4 3 (2n)! 3 (2π)2n  n−1 ∞ k−1 X X (−1) · k ζ (2k + 1) (2k)! ζ (2k)  +  (n ∈ N)   + (2n − 2k)! 2 2k (2n + 2k)! 32k k=1 k=0 3π (25) " 2n 2(2π) 22n−3 (22n − 1)π B2n ζ (2n + 1) = (−1)n (2n)! n(24n+1 + 22n − 1)

+

+

  X n−1 (−1)n π 1 (−1)k−1 · k ζ (2k + 1) ζ 2n, +   2n 4 (2n − 2k)! 1 2k 2(2π) k=1 2π ∞ X k=0

(2k)! ζ (2k) (2n + 2k)! 42k

# (n ∈ N),

(26)

Evaluations and Series Representations

421

and " 2(2π)2n 22n−3 (32n − 1)π B2n ζ (2n + 1) = (−1) √ 2n 2n 2n n(6 + 3 + 2 − 1) 3 (2n)!    (−1)n π 1 1
+ √ + ζ 2n, ζ 2n, 3 6 2 3 (2π)2n n

 n−1 ∞ k−1 X X (−1) · k ζ (2k + 1) (2k)! ζ (2k)  + + 
(2n − 2k)! 1 π 2k (2n + 2k)! 62k k=1

3

(27)

(n ∈ N),

k=0

in terms of the Bernoulli numbers Bn , defined by 1.7(2). By comparing the series representation (26) with a known result, due to Srivastava [1083, p. 9, Eq. (41)] (see also Srivastava [1082, p. 5, Eq. (3.3)]), Srivastava and Tsumura [1111] derived an interesting identity involving the Zeta functions ζ (s) and ζ (s, a) (and their derivatives with respect to s), which is given below:  n  o B (−1)n π 1 2n = (22n−2 − 1) log 2 − 22n−3 (22n − 1)π ζ 2n, 2n 4 (2n)! 2(2π)   (28) 1 22n−1 − 1 0 42n−1 0 − ζ (1 − 2n) − ζ 1 − 2n, (n ∈ N). 2(2n − 1)! (2n − 1)! 4 By applying (25), (26) and (27) and Srivastava’s series representations 4.2(21), 4.2(22) and 4.2(23), Srivastava and Tsumura [1111] proved their main results given by     2n H2n+1 − log 2 π 3 (32n+2 − 1)π 2π  + √ ζ (2n + 1) = (−1)n−1 B2n+2 3 (2n + 1)! 2 3 (2n + 2)!   X n−1 1 (−1)k ζ (2k+1) ζ 2n+2, + +√  2k 2n+1 3 (2n−2k+1)! 3 (2π) 2 k=1 3π # ∞ X (2k − 1)! ζ (2k) +2 (n ∈ N), (2n + 2k + 1)! 32k (−1)n−1

(29)

k=1

   π 2n H2n + 1 − log 12 π 22n (22n + 2 − 1)π  ζ (2n+1) = (−1)n − 1 + B2n + 2 2 (2n + 1)! (2n + 2)!   nX −1 ( − 1)n − 1 1 ( − 1)k ζ (2k + 1) ζ 2n + 2, +   2n + 1 4 (2n − 2k + 1)! 1 2k 2(2π) k=1 2π  ∞ X (2k − 1)! ζ (2k)  +2 (n ∈ N), (2n + 2k + 1)! 42k

+

k=1

(30)

422

Zeta and q-Zeta Functions and Associated Series and Integrals

and 

H2n+1 − log



1 3π



22n (32n+2 − 1)π + √ B2n+2 3 (2n + 1)! 3 (2n + 2)!      1 1 (−1)n−1 ζ 2n + 2, + ζ 2n + 2, + √ 3 6 2 3 (2π)2n+1 (31) n−1 X (−1)k ζ (2k + 1) +   (2n − 2k + 1)! 1 2k k=1 3π  ∞ X (2k − 1)!(2n + 2k + 1)! ζ (2k)  +2 (n ∈ N). 62k

ζ (2n + 1) = (−1)n−1

 π 2n



k=1

Each of the main series representations (29), (30) and (31) belongs to the class containing Wilton’s formula (1) and Srivastava’s formula (2). If, for convenience, we denote the summands of the infinite series in the representations (1), (2), (29), (30) (j) and (31), by S k (j = 1, 2, 3, 4, 5), respectively, and apply Stirling’s formula 1.1(34) and the fact that ζ (2k) → 1 as k → ∞, we easily obtain the following order estimates (cf. Equations (3) and (4)):
(j) S k = O k−2n−2 · m−2k (k → ∞; n ∈ N) (32) (m = 2 when j = 1; m = 3 when j = 3; m = 4 when j = 2 and j = 4; m = 6 when j = 5). Clearly, therefore, of these five series representations for ζ (2n + 1), the result (31) involves the most rapidly convergent series. Conversely, the rate of convergence of the series involved in each of the Srivastava-Tsumura results (29), (30) and (31) is obviously much better than that of the series involved in Wilton’s result (1). And, in particular, the rate of convergence of the series involved in the Srivastava-Tsumura result (30) is as good as that of the series involved in Srivastava’s result (2).

4.4 Computational Results Srivastava [1086] investigated rather systematically several interesting evaluations and representations of ζ (s) when s ∈ N \ {1}. In one of many computationally useful special cases considered by him, it is observed that ζ (3) can be represented by means of a series which converges much more rapidly than that in Euler’s celebrated formula 3.1(8), as well as the series 4.2(2) used recently by Ape´ ry [56] in his proof of the irrationality of ζ (3). Symbolic and numeric computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of the series (38) below are capable of producing an accuracy of seven decimal places. Since √
2 (1) i cot iz = coth z = 2z + 1 i := −1 , e −1

Evaluations and Series Representations

423

by replacing z in the known expansion 4.2(4) by 21 iπz, it is easily seen that (cf., e.g., Koblitz [681, p. 25]; see also Erde´ lyi et al. [421, p. 51, Eq. 1.20(1)]) ∞

πz πz X (−1)k+1 ζ (2k) 2k + = z eπ z − 1 2 22k−1

(|z| < 2) .

(2)

k=0

By setting z = it in (2), multiplying both sides by tm−1 (m ∈ N) and then integrating the resulting equation from t = 0 to t = τ (0 < τ < 2), if we apply the (readily derivable) integral formula:   Zτ m m m!eλτ j X (−1) (−λτ )  tm eλt dt = − e−λτ  (λ 6= 0), (3) j! λm+1 j=0

0

we obtain )  m (X ∞ i cos(kπτ ) + i sin(kπτ ) m! − ζ (m + 1) πτ km+1 k=1 ! ∞ ∞ X X πτ cos(kπτ ) sin(kπτ ) +i + + k 2(m + 1) k k=1

k=1

+

m−1 X j=1

m! (m − j)!



i πτ

∞ X ζ (2k)  τ 2k = k + 21 m 2

j X ∞ k=1

cos(kπτ ) + i sin(kπτ ) kj+1

(4)

(m ∈ N; 0 < τ < 2),

k=0

which, in the special case when τ = 1, would immediately simplify to the form:    i m iπ m+1 − 2 −1 m! ζ (m + 1) 2(m + 1) 2π   m−1 X m!
i j j 2 −1 ζ (j + 1) = log 2 + (m − j)! 2π j=1

+

∞ X

ζ (2k)   1 2k k=0 k + 2 m 2

(5)

(m ∈ N) ,

where we have also applied the last part of the definition 2.3(1). Furthermore, in (4), (5) and elsewhere in this section, an empty sum is interpreted (as usual) to be nil. Now, for a given sequence {n }∞ n=1 , it is easily verified that m−1 X j=1

j =

[(m−1)/2] X j=1

2j +

[m/2] X j=1

2j−1 ,

(6)

424

Zeta and q-Zeta Functions and Associated Series and Integrals

where, just as in 4.1(17), [κ] denotes the greatest integer in κ. Thus, we find from (5) that    i m iπ − 2m+1 − 1 m! ζ (m + 1) 2(m + 1) 2π
  [(m−1)/2] 2j X j m (2j)! 2 − 1 = log 2 + (−1) ζ (2j + 1) 2j (2π)2j j=1

+i

[m/2] X

(−1)

j−1

j=1

+




 (2j − 1)! 22j−1 − 1 m ζ (2j) 2j − 1 (2π)2j−1

∞ X

ζ (2k)   1 2k k=0 k + 2 m 2

(7)

(m ∈ N).

Setting m = 2n (n ∈ N) in (7) and then equating the real and imaginary parts in the resulting equation, we obtain (Srivastava [1086]) ζ (2n + 1) = (−1)n−1

(2π)2n
(2n)! 22n+1 − 1


  ∞ n−1 2j −1 X X (2j)! 2 ζ (2k) 2n  ζ (2j + 1) + · log 2 + (−1) j 2j (2π)2j (k + n)22k 

j=1

(n ∈ N)

k=0

(8) and ζ (2n) = (−1)n−1

(2π)2n−1 (2n)!(22n−1 − 1)




  n−1 2j−1 −1 X (2j−1)! 2 2n π · + (−1) j ζ (2j) 2(2n + 1) 2j−1 (2π)2j−1

(n ∈ N).

j=1

(9) Similarly, if we set m = 2n + 1 (n ∈ N0 ) in (7), we shall obtain the following results, which can provide series representations for ζ (2n + 1) and ζ (2n) when n ∈ N :
  n 2j X j−1 2n + 1 (2j)! 2 − 1 (−1) ζ (2j + 1) 2j (2π)2j j=1 (10) ∞ X ζ (2k)   = log 2 + (n ∈ N0 ) 1 2k k=0 k + n + 2 2

Evaluations and Series Representations

425

and (2π)2n−1
(2n − 1)! 22n − 1  
  n−1 2j−1 − 1 X (2j − 1)! 2 2n − 1 π ζ (2j) (−1) j · + 2j − 1 4n (2π)2j−1

ζ (2n) = (−1)n−1

(n ∈ N).

j=1

(11) Indeed, in its particular cases when n = 0 and n ∈ N, the summation formula (10) immediately yields the known sum 3.4(493) and (Srivastava [1086]) (2π)2n
(2n + 1)! 22n − 1 
  n−1 2j X j 2n + 1 (2j)! 2 − 1  · log 2 + ζ (2j + 1) (−1) 2j (2π)2j j=1  ∞ X ζ (2k)  (n ∈ N),   + 1 2k k + n + 2 k=0 2

ζ (2n + 1) = (−1)n−1

(12)

respectively. Each of the recursion formulas (9) and (11) can be used to evaluate ζ (2n) (n ∈ N). Formula (9) was proven, in a markedly different way, by (for example) Stark [1121, p. 199, Eq. (5)]. Formulas (9) and (11), together, yield the series identity (Srivastava [1086]):
   n−1 2j−1 − 1 X π (2n − 1)22n−1 − 2n j−1 2n − 1 (2j − 1)! 2
ζ (2j) = (−1) 2j − 1 (2π)2j−1 4n(2n + 1) 22n−1 − 1 j=1
  n−1 X 22n − 1 2n (2j − 1)! 22j−1 − 1 j−1
+ (−1) ζ (2j) (n ∈ N) 2j − 1 (2π)2j−1 2n 22n−1 − 1 j=1

(13) or, equivalently,   n−1   o (2j − 2)! 22j−1 − 1
X 2n − 1 n (−1) j−1 2(n − j) 22n−1 − 1 − 22n−1 ζ (2j) 2j − 2 (2π)2j−1 j=1  π (2n − 1)22n−1 − 2n = (n ∈ N), (14) 4n(2n + 1) which can also be used as a recurrence relation for evaluating ζ (2n) (n ∈ N).

426

Zeta and q-Zeta Functions and Associated Series and Integrals

An integral representation for ζ (2n + 1), which is equivalent to the series representation (8), was given earlier by Da¸browski [362, p. 203, Eq. (16); p. 206], who also mentioned the existence of (but did not fully state) the series representation (12). The series representation (8) is derived also in a forthcoming paper by Borwein et al. [153, Eq. (57)]. For n = 1, (8) immediately yields the series representation 4.2(5) which, in conjunction with the known sum 3.4(493), would lead us readily to Euler’s formula 3.1(8). Conversely, by setting n = 1 in the series representation (12), we obtain 4π 2 ζ (3) = 9

! ∞ X 1 ζ (2k) log 2 + , 2 (2k + 3)22k

(15)

k=0

which was derived independently by (for example) Glasser [491, p. 446, Eq. (12)], Zhang and Williams [1251, p. 1585, Eq. (2.13)] and Da¸browski [362, p. 206] (see also Chen and Srivastava [252, p. 183, Eq. (2.15)]). By suitably combining the series occurring in 4.2(5), 3.4(493) and (15), it is not difficult to derive several other series representations for ζ (3), which are analogous to Euler’s formula 3.1(8). More generally, since λk2 + µk + ν (2k + 2n − 1)(2k + 2n)(2k + 2n + 1)

(16)

A B C = + + , 2k + 2n − 1 2k + 2n 2k + 2n + 1

where, for convenience,   1 1 λn2 − (λ + µ)n + (λ + 2µ + 4ν) , 2 4   B = Bn (λ, µ, ν) := − λn2 − µn + ν ,

A = An (λ, µ, ν) :=

(17) (18)

and   1 1 2 C = Cn (λ, µ, ν) := λn + (λ − µ)n + (λ − 2µ + 4ν) , 2 4

(19)

by appealing to (8), (10) with n replaced by n − 1 and (12), we can derive the following unification of a large number of known (or new) series representations, including (for

Evaluations and Series Representations

427

example) Euler’s result 3.1(8) (Srivastava [1086]): (−1)n−1 (2π)2n

(2n)! B + (2n + 1) 22n − 1 C    n−1 X 1 2n − 1 ·  λ log 2 + 2j(2j − 1)A + [λ(4n − 1) − 2µ] nj (−1) j 4 2j − 2

ζ (2n + 1) =



22n+1 − 1

j=1


  (2j − 2)! 22j − 1 1 + λn n + ζ (2j + 1) 2 (2π)2j 
∞ X λk2 + µk + ν ζ (2k)  + (2k + 2n − 1)(k + n)(2k + 2n + 1)22k

(20)

(n ∈ N; λ, µ, ν ∈ C),

k=0

where A, B and C are given by (17), (18) and (19), respectively. For λ = 0, the series representation (20) simplifies to the form: ζ (2n + 1) =

n

(−1)n−1 (2π)2n h io
 22n+1 − 1 (µn − ν) − 22n − 1 n + 12 µ(n + 21 ) − ν

(2n)!        n−1 X 1 j 2n − 1  − 2µnj · j(2j − 1) ν − µ n − (−1) 2 2j − 2


j=1


∞ X (2j−2)! 22j −1 (µk+ν)ζ (2k)  · ζ (2j+1) + (2π)2j (2k+2n−1)(k+n)(2k+2n+1)22k k=0

(n ∈ N; µ, ν ∈ C).

(21)

Furthermore, by setting λ=µ=0

and ν = 1

in (20) or (alternatively) by setting µ=0

and ν = 1

in (21), we immediately obtain the series representation (Srivastava [1086]): ζ (2n + 1) 
  n−1 X (−1)n−1 (2π)2n 2n − 1 (2j)! 22j − 1 j   = (−1) ζ (2j + 1) 2j − 2 (2π)2j (2n)! 22n (2n − 3) − 2n + 1 j=1 (22)  ∞ X ζ (2k)  (n ∈ N), +2 (2k + 2n − 1)(k + n)(2k + 2n + 1)22k k=0

428

Zeta and q-Zeta Functions and Associated Series and Integrals

which, in the special case when n = 1, was given by Chen and Srivastava [252, p. 189, Eq. (2.45)]. Of the three representations (20), (21) and (22) for ζ (2n + 1) (n ∈ N), the infinite series in (22) converges most rapidly. Many other families of rapidly convergent series representations for ζ (2n + 1) (n ∈ N) can be found in the recent works of Srivastava [1083, 1084] (and, indeed, also in the numerous references already cited in each of these earlier works), including the results presented in previous sections (especially Section 4.2). For various suitable special values of the parameters λ, µ, and ν, we can easily deduce from (20) and (21) several known (or new) series representations for ζ (2n + 1) (n ∈ N). For example, if we set µ=2

and ν = 2n + 1

in our series representation (21), we shall obtain (Srivastava [1086])    n−1 2n X 2n − 1 (2π)
 (−1) j ζ (2n + 1) = (−1)n−1 2j − 1 (2n)! 22n+1 − 1 j=1 #
∞ X (2j)! 22j − 1 ζ (2k) · ζ (2j + 1) − (2π)2j (2k + 2n − 1)(k + n)22k

(23) (n ∈ N),

k=0

which, in the special case when n = 1, immediately yields Euler’s formula 3.1(8). The following additional series representations for ζ (2n + 1) (n ∈ N), which are analogous to (23), can also be deduced similarly from (21) (Srivastava [1086]):

ζ (2n + 1) = (−1)n−1



(2π)2n

2n

(2n)! (2n − 1)22n − 2n 

  n−1 X 2n − 1 (−1) j 2j − 2 j=1 #


∞ X (2j)! 22j − 1 ζ (2k) · ζ (2j + 1) − (2π)2j (k + n)(2k + 2n + 1)22k

(24)

(n ∈ N)

k=0

and     n−1 X 4nj − 2j + 1 2n − 1
 (−1) j ζ (2n + 1) = (−1)n−1 2j − 1 2j − 2 (2n + 1)! 22n − 1 j=1 #
∞ X (2j)! 22j − 1 ζ (2k) · ζ (2j + 1) − 4 (n ∈ N). (2π)2j (2k + 2n − 1)(2k + 2n + 1)22k (2π)2n

k=0

(25)

Evaluations and Series Representations

429

The special case of each of the last two series representations (24) and (25) when n = 1 was given by Zhang and Williams [1251, p. 1586]. For n = 2, Srivastava’s formulas (22) to (25) would readily yield the following series representations for ζ (5): ∞

ζ (5) = ζ (5) =

8π 4 X π2 ζ (2k) ζ (3) − , 13 13 (2k + 3)(2k + 4)(2k + 5)22k k=0 ∞ 4X

3π 2 4π ζ (3) + 31 93

k=0 ∞

ζ (2k) , (2k + 3)(2k + 4)22k

π4 X ζ (2k) π2 ζ (3) + ζ (5) = , 11 33 (2k + 4)(2k + 5)22k

(26)

(27)

(28)

k=0

and ∞

ζ (5) =

7π 2 8π 4 X ζ (2k) ζ (3) + . 75 225 (2k + 3)(2k + 5)22k

(29)

k=0

Next, with a view to further improving the rate of convergence in the most rapidly convergent series representations (22) and (26) considered in this work, we observe that 1 (2k + 2n − 1)(2k + 2n)(2k + 2n + 1)(2k + 2n + 2)   1 1 1 1 1 = − − . 6 2k + 2n − 1 2k + 2n + 2 2 (2k + 2n)(2k + 2n + 1)

(30)

Thus, by applying the series representations (10) with n replaced by n − 1, (8) with n replaced by n + 1 and (24), we obtain (Srivastava [1086]) 
2π 2 22n+2 + n(2n − 3) 22n − 1 − 1
ζ (2n + 1) ζ (2n + 3) = (n + 1)(2n + 1) 22n+3 − 1    n−1 2n+2 X (2π) 2n − 1 n−1 j 
+ (−1) (−1) 2j (2n + 2)! 22n+3 − 1 j=1
    (2j)! 22j − 1 2n + 2 2n − 1 − + 6n ζ (2j + 1) 2j 2j − 2 (2π)2j + 12

∞ X k=0

 ζ (2k)  (2k + 2n − 1)(2k + 2n)(2k + 2n + 1)(2k + 2n + 2)22k

where the series converges faster than that in (22).

(31)

(n ∈ N),

430

Zeta and q-Zeta Functions and Associated Series and Integrals

In its special case when n = 1, (31) readily yields the following improved version of the series representation (26) above (cf. Zhang and Williams [1251, p. 1590, Eq. (3.14)]: ∞

ζ (5) =

8π 4 X ζ (2k) 4π 2 ζ (3) + 31 31 (2k + 1)(2k + 2)(2k + 3)(2k + 4)22k

(32)

k=0

in which ζ (3) can be replaced by its known value −4π 2 ζ 0 (−2) given by 2.3(22) for n = 1. Yet another rapidly convergent series representation for ζ (2n + 3) (n ∈ N), analogous to (31), can be derived by means of the identity: 1 (2k + 2n)(2k + 2n + 1)(2k + 2n + 2)(2k + 2n + 3)   1 1 1 1 1 = − − , 6 2k + 2n 2k + 2n + 3 2 (2k + 2n + 1)(2k + 2n + 2)

(33)

together with the series representations (8), (12) with n replaced by n + 1 and (23) with n replaced by n + 1. We, thus, obtain the series representation (Srivastava [1086]): o n

2π 2 13 (2n + 1) 2n2 − 4n + 3 22n − 1 − 22n+1 + 1  ζ (2n + 1) ζ (2n + 3) = (n + 1)(2n + 1) (2n − 3)22n+2 − 2n    n−1 2n+2 X (2π) 2n   (−1) j + (−1)n−1 2j (2n + 2)! (2n − 3)22n+2 − 2n j=1
    (2j)! 22j − 1 2n + 3 2n + 1 ζ (2j + 1) − +3 2j 2j − 1 (2π)2j # ∞ X ζ (2k) + 12 (n ∈ N), (2k + 2n)(2k + 2n + 1)(2k + 2n + 2)(2k + 2n + 3)22k k=0

(34) which, in the special case when n = 1, yields ∞

ζ (5) =

4π 4 X ζ (2k) 2π 2 ζ (3) − , 27 9 (2k + 2)(2k + 3)(2k + 4)(2k + 5)22k

(35)

k=0

where the series obviously converges faster than that in (26). Lastly, by applying the identity: 1 1 1 = 2k(2k + 2n − 1)(2k + 2n)(2k + 2n + 1) 2n(2n − 1)(2n + 1) 2k 1 1 1 1 1 1 − + − 2(2n − 1) 2k + 2n − 1 2n 2k + 2n 2(2n + 1) 2k + 2n + 1

(36)

Evaluations and Series Representations

431

in conjunction with the series representations (10) with n replaced by n − 1, (8), (12) and the known result 3.4(26), we arrive at the following series representation for ζ (2n + 1) (n ∈ N) (Srivastava [1086]): " 12n2 − 1 (2π)2n n−1  ζ (2n + 1) = (−1)
2 · (2n − 1)! n − 1 − (n − 2)22n 2n2 4n2 − 1 2
  n−1 2j X log π j 2n − 2 (2j − 1)! 2 − 1
− (−1) ζ (2j + 1) − 2j − 3 (2π)2j n 4n2 − 1 j=2 # ∞ X ζ (2k) (n ∈ N), + k(k + n)(2k + 2n − 1)(2k + 2n + 1)22k k=1

(37) where we have also applied the fact that ζ (0) = − 12 . For n = 1, (37) reduces immediately to Wilton’s formula (cf. Hansen [531, p. 357, Entry (54.5.9)]; see also Chen and Srivastava [252, p. 181, Eq. (2.1)]): ! ∞ X π 2 11 1 ζ (2k) ζ (3) = − log π + . (38) 2 18 3 k(k + 1)(2k + 1)(2k + 3)22k k=1

Furthermore, in its special case when n = 2, (37) would yield the following companion of the series representations (32) and (35): ! ∞ X 2π 4 47 ζ (2k) ζ (5) = log π − − 30 , (39) 45 60 k(k + 2)(2k + 3)(2k + 5)22k k=1

which does not contain a term involving ζ (3) on the right-hand side. By eliminating ζ (2n + 3) between the results (31) and (34), we can obtain a series representation for ζ (2n + 1) (n ∈ N), which would converge as rapidly as the series in (37). For n ∈ N, we, thus, find that (Srivastava [1086])  n n−1 o 2n (2π) X ζ (2n + 1) = (−1)n−1 (−1) j (2n − 3)22n+2 − 2n (2n)!1n j=1

        2n − 1 2n + 2 2n − 1 − + 6n − 22n+3 − 1 2j 2j 2j − 2
      (2j)! 22j − 1 2n 2n + 3 2n + 1 ζ (2j + 1) · − +3 2j 2j 2j − 1 (2π)2j + 12

∞ X k=0

(40)

# (ξn k + ηn ) ζ (2k) , (2k + 2n − 1)(2k + 2n)(2k + 2n + 1)(2k + 2n + 2)(2k + 2n + 3)22k

432

Zeta and q-Zeta Functions and Associated Series and Integrals

where, for convenience,   1    2n+3 2 2n 2n+1 1n := 2 −1 (2n + 1) 2n − 4n + 3 2 − 1 − 2 +1 3 n on   o − (2n − 3)22n+2 − 2n 22n+2 + n(2n − 3) 22n − 1 − 1 , n o ξn := 2 (2n − 5)22n+2 − 2n + 1 ,

(41)

(42)

and   ηn := 4n2 − 4n − 7 22n+2 − (2n + 1)2 .

(43)

In its special case when n = 1, (40) yields the following series representation (Srivastava [1086]): ∞

ζ (3) = −

6π 2 X (98k + 121)ζ (2k) , 23 (2k + 1)(2k + 2)(2k + 3)(2k + 4)(2k + 5)22k

(44)

k=0

where the series obviously converges much more rapidly than that in each of the celebrated results 3.1(8) and 4.2(2). We conclude this section by summarizing below the results of some symbolic and numeric computations with the series in (44) using Mathematica (Version 4.0) for Linux: In[1] := (98k + 121)Zeta[2k] / ((2k + 1)(2k + 2)(2k + 3)(2k + 4)(2k + 5)2(2k)) (121 + 98k)Zeta[2k] Out[1] = 2k 2 (1 + 2k)(2 + 2k)(3 + 2k)(4 + 2k)(5 + 2k) In[2] := Sum [%, {k, 1, Infinity}] //Simplify 121 23 Zeta[3] Out[2] = − 240 6 Pi2 In[3] := N[%] Out[3] = 0.0372903 In[4] := Sum [N[%1]//Evaluate, {k, 1, 50}] Out[4] = 0.0372903 In[5] := N Sum [%1//Evaluate, {k, 1, Infinity}] Out [5] = 0.0372903 Since ζ (0) = − 12 , Out [2] evidently validates the series representation (44) symbolically. Furthermore, our numeric computations in Out [3], Out [4] and Out [5], together, exhibit the fact that only 50 terms (k = 1 to k = 50) of the series in (44) can produce an accuracy of seven decimal places (see, for details, Srivastava [1086]).

Evaluations and Series Representations

433

Problems 1. Prove that ∞ ∞ ∞ X X X 1 Hn Hn = 2 = 2 , n2 n3 (n + 1)2 n=1

n=1

n=1

where Hn denotes the harmonic numbers, defined by 3.2(36). (Cf. Equations 2.3(54) and 3.5(18); see also Klamkin [669, p. 195]) 2. Prove that ζ (4) =

∞ 36 X 1
. 4 2k 17 k k=1 k

(Comtet [337, p. 89]; Cohen [334, p. 280]) 3. Prove that π2 ζ (3) = 6

(

∞

3 1 π X ζ (2k) − log + 4 2 3 k(2k + 1)(2k + 2) 62k

) .

k=1

(Zhang and Williams [1251, Section 2]; Chen and Srivastava [252, p. 184]) 4. Prove that π2 ζ (3) = 14

(

∞

X (22k−1 − 1)ζ (2k) 3 π − log − 4 2 4 k(2k + 1)(2k + 2) 24k

) .

k=1

(Ewell [438, p. 1011]; Chen and Srivastava [252, p. 187]) 5. Prove the following series representations for ζ (3): 2π 2 ζ (3) = 35 ζ (3) =

2π 2 65

ζ (3) =

4π 2 25

ζ (3) =

4π 2 31

) ∞ X 3 π (22k−1 − 1)ζ (2k) − log − 4 ; 2 6 k(2k + 1)(2k + 2) 62k k=1 ( ! ) ∞ X π2 (32k−1 − 1)ζ (2k) 3 − log −6 ; 27 k(2k + 1)(2k + 2) 62k k=1 ( )   ∞ X 3 1 8π (32k−1 − 22k−1 )ζ (2k) − log −6 ; 4 2 27 k(2k + 1)(2k + 2) 62k k=1 ( )   ∞ X 3 1 27π (42k−1 − 32k−1 )ζ (2k) − log − 12 ; 4 2 128 k(2k + 1)(2k + 2) 122k (

k=1

and 4π 2 ζ (3) = 37

(

)   ∞ X 3 1 4π (32k−1 − 22k−1 )ζ (2k) − log −6 . 4 2 27 k(2k + 1)(2k + 2) 122k k=1

(Chen and Srivastava [252, p. 187])

434

Zeta and q-Zeta Functions and Associated Series and Integrals

6. For n ∈ N \ {1}, show that ζ (2n + 1) =

(n−1 (−1)n (2π )2n X 22k
2n+1 2 − 1 n k=1 (2k − 1)! (2n − 2k)! (2k + 1)2 +

n−1 X k=1

·

∞

X (−1) j (2j − 1)! 22k (2k − 1)! (2n − 2k)! 22j−1 π 2j j=1

 min(2j−2,2k) X i=0

2k i



 ∞ X (2k)! ζ (2k) + . 2j − i − 1 (2k + 4)! 22k

(−1)i ζ (2j)

k=0

(Zhang and Williams [1251, p. 1591]) 7. Let

Cp,q :=

j−1 ∞ X X 1 jp k q

(p ∈ N \ {1}; q ∈ N)

j=2 k=1

and

Cp,q,r :=

j−1 X ∞ X k−1 X j=3 k=2 i=1

1 jp kq ir

(p ∈ N \ {1}; q, r ∈ N).

Show that  p−2 q+n−1 X X (q)n (r)m Cp−n,r+m,q+n−m Cp,q,r = (−1)q  n!m! n=0 m=0 ) r−1 X (q)n+m Cp−n,q+n+m,r−m + n!m! m=0

+

q−2 X n=0

·

( r−2 X

(p)n (p + q − 2)! (−1)n ζ (p + n) Cq−n,r + (−1)q−1 n! (p − 1)! (q − 1)!   (−1)m ζ (r − m) Cp+q−1,m+1 + ζ (p + q + m)

m=0

   + (−1)r−1 Cp+q−1,r,1 + Cp+q+r−1,1 + Cp+q−1,r+1 + ζ (p + q + r) , where empty sums are interpreted (as usual) to be nil. (Cf. Berndt [122, p. 253]; Markett [801, p. 122]) 8. Prove that ζ (2n + 1) = (−1)n

∞ 4(2π )2n X R2n+1,k ζ (2k) (2n + 1)! k=0

(n ∈ N),

Evaluations and Series Representations

435

where the coefficients R2n+1,k are rational and given by R2n+1,k :=

2n   X 2n

m

m=0

(2n + 1) B2n−m 2k+m+1 2 (2k + m + 1) (m + 1)

(k ∈ N0 )

and by the family of generating functions ∞ X

R2n+1,k

n=0

t2n+1 (2n + 1)!

(2k − 1)! = 2k−1 t t (e − 1)

1−e

1 2t

2k−1 X m=0

(−1)m m!

 m ! t 2

(k ∈ N; |t| < 2π).

(Cvijovi´c and Klinowski [351, p. 1265]; see also 4.2(39) et seq.) 9. Prove that ζ (n) =

n−2 X ∞ X

1

j=1 p, q=1

pj (p + q)n−j

(n ∈ N \ {1, 2}).

(cf. Briggs [177]; see also Markett [801, p. 115]) 10. Let a, b, c be real numbers greater than 1 and P(a, b, c) =

∞ X

r−a

r=1

r X

k−b

k=1

k X

l−c .

l=1

Then show that P(a, b, c) + P(a, c, b) + P(b, c, a) + P(b, a, c) + P(c, a, b) + P(c, b, a) = ζ (a) ζ (b) ζ (c) + ζ (a) ζ (b + c) + ζ (b) ζ (c + a) + ζ (c) ζ (a + b) + 2 ζ (a + b + c), where ζ is the Riemann Zeta function, defined by 2.3(1). (Sitaramachandrarao and Subbarao [1037, p. 471]) 11. For each pair (m, n) ∈ N0 × N, one defines A2m (n) as follows: (i) A2m (1) := B2m , and (ii) for n ∈ N \ {1}, A2m (n) :=

2m
2j1 , ..., 2jn

X

{2j1 + 1} {2 (j1 + j2 ) + 1} · · · {2 (j1 + · · · + jn−1 ) + 1}

· B2j1 · · · B2jn ,

where the
sum is taken over all (j1 , . . . , jn ) ∈ N0 n , such that j1 + · · · + jn = m and 2m 2j1 , ..., 2jn is a multinomial coefficient; Bj are Bernoulli numbers, defined by 1.6(2). Then show that ∞

ζ (n) =

2n−2 2 X π 2k π (−1)k A2k (n − 2) n 2 −1 (2k + 2)!

(n ∈ N \ {1, 2}).

k=0

(Ewell [437, p. 61])

436

Zeta and q-Zeta Functions and Associated Series and Integrals

12. Show that ζ (3) =

  ∞ X 1 2 3 π −4 e−2πn σ−3 (n) 2 π 2 n2 + π n + 45 2 n=1

and ∞

ζ (3) =

X 7 3 π −2 e−2π n σ−3 (n), 180 n=1

where σν (n) is given in Problem 20 of Chapter 2. (Cf. Terras [1147, p. 181] and Grosswald [514]; see also Problem 21 of Chapter 2) 13. Consider the cosecant integral: Is (ω) :=

π/ω Z ts csc2 t dt

(<(s) > 1;

ω > 1) .

0

Prove the following identities: Is (ω) = −

Is (ω) = −

 π s ω  π s ω

cot

cot

π  ω π  ω

π/ω Z +s ts−1 cot t dt

(<(s) > 1;

ω > 1) ;

0

− 2s

∞  π s−1 X

ω

k=0

ζ (2k) (s + 2k − 1)ω2k (<(s) > 1;

Zπ/2 ∞  π s−1 X ζ (2k) ts csc2 t dt = −2s 2 (s + 2k − 1)22k

ω > 1) ;

(<(s) > 1) ;

k=0

0

Zπ/4 ∞  π s−1 X  π s ζ (2k) − 2s ts csc2 t dt = − 4 4 (s + 2k − 1)42k

(<(s) > 1) ;

k=0

0

Zπ/3 ∞  π s−1 X ζ (2k) 1  π s ts csc2 t dt = − √ − 2s 3 (s + 2k − 1)32k 3 3 k=0

(<(s) > 1) ;

0

Zπ/6 ∞  π s−1 X √  π s−1 ζ (2k) ts csc2 t dt = − 3 − 2s 6 6 (s + 2k − 1)62k

(<(s) > 1) ;

k=0

0

Is (ω) = −

∞  π s h  π  i X γ (s, −2kπ i/ω) cot + i −2is ω ω (−2ki)s

(<(s) > 1; ω > 1) ,

k=1

where γ (z, α) denotes the incomplete Gamma function, defined by 1.1(76);   m j X α  (m ∈ N0 ) ; γ (m + 1, α) = m! 1 − e−α j! j=0

Evaluations and Series Representations

I2n (ω) = −

 π 2n ω

cot

437

π  ω

− (2n)!

n−1 X

 π 2n−1 ω

"

∞

X cos(2kπ/ω) 1 (2n − 1)! k k=1

∞

 ω 2j X cos(2kπ/ω) (−1) j + (2n − 2j − 1)! 2π k2j+1 j=1 k=1  n ∞ X (−1) j  ω 2j−1 X sin(2kπ/ω)  + (2n − 2j)! 2π k2j j=1

(n ∈ N; ω > 1) ;

k=1

 π 2n+1

π 

(2n + 1)! ζ (2n + 1) 22n ∞ n  π 2n X cos (2kπ/ω)) X (−1) j  ω 2j  1 − (2n + 1)! + ω (2n)! k (2n − 2j)! 2π j=1 k=1  ∞ n ∞  ω 2j−1 X X cos(2kπ/ω) X (−1) j sin(2kπ/ω)  · + (2n − 2j + 1)! 2π k2j+1 k2j

I2n+1 (ω) = −

ω

cot

ω

j=1

k=1

∞ X k=0

ζ (2k) (2k + 2n − 1)ω2k

+ (−1)n

k=1

(n ∈ N; ω > 1) ; h  i π 1 = − log 2 sin 2 ω    n−1  ω 2j (2n − 1)! X (−1) j 2π + C`2j+1 2 (2n − 2j − 1)! 2π ω j=1    n X 2π  (−1) j  ω 2j−1 + C`2j (2n − 2j)! 2π ω j=1

and ∞ X

 2n h  π i ζ (2k) 1 n−1 (2n)! ω = (−1) ζ (2n + 1) − log 2 sin 2 ω 22n+1 π (2k + 2n)ω2k k=0 n  ω 2j (2n)! X (−1) j + 2 (2n − 2j + 1)! 2π j=1      2π 2π 2π · (2n − 2j + 1)C`2j+1 + C`2j , ω ω ω

in terms of the generalized Clausen functions C`2n and C`2n+1 , defined by 2.4(80) and 2.4(81) and n ∈ N and ω > 1. (Srivastava, Glasser, and Adamchik [1097]) 14. Continuing Problem 13, show that  ∞ X (2π )2n ζ (2k) n−1
 log 2 + 2 ζ (2n + 1) = (−1) (2k + 2n + 1)22k (2n + 1)! 22n − 1 k=0  ! n−1 j 2j X (−1) 2 −1 + (2n + 1)! ζ (2j + 1) (2n − 2j + 1)! (2π)2j j=1

438

Zeta and q-Zeta Functions and Associated Series and Integrals

and  ∞ 2n X (2π ) ζ 2k)
log 2 + ζ (2n + 1) = (−1)n−1 2n+1 (k + n)22k (2n)! 2 −1 k=0  ! n−1 X (−1) j 22j − 1 + (2n)! ζ (2j + 1) , (2n − 2j)! (2π )2j j=1

for n ∈ N, respectively; ∞ X sin(kx + y) k=1

ks

=

    1 x  (2π )s csc(π s) cos y − π s ζ 1 − s, 20(s) 2 2π      x 1 , − cos y + π s ζ 1 − s, 1 − 2 2π

for <(s) > 1 and 0 < x < 2π; ∞ X sin(2kπ/ω) k=1

ks

=

      (2π )s 1 1 1 csc πs ζ 1 − s, − ζ 1 − s, 1 − , 40(s) 2 ω ω

for <(s) > 1 and ω > 1;  ∞ 2n X ζ (2k) (2π )
log 3 + 4 ζ (2n + 1) = (−1)n−1 2n (2k + 2n + 1)32k (2n + 1)! 3 − 1 k=0 ! n−1 X (−1) j 32j − 1 + (2n + 1)! ζ (2j + 1) (2n − 2j + 1)! (2π )2j j=1   
n+1 2ζ 2j, 31 − 32j − 1 ζ (2j) (2n + 1)! X (−1) j  − √ (2n − 2j + 2)! (2π)2j−1 3

(n ∈ N)

j=1

and " ∞ X (2π )2n ζ (2k)
log 3 + 2 ζ (2n + 1) = (−1) (k + n)32k (2n)! 32n+1 − 1 k=0 ! n−1 X (−1) j 32j − 1 + (2n)! ζ (2j + 1) (2n − 2j)! (2π )2j j=1   
n 2ζ 2j, 13 − 32j − 1 ζ (2j) (2n)! X (−1) j  − √ (2n − 2j + 1)! (2π)2j−1 3 n−1

j=1

" ∞ X π2 ζ (2k) log 3 + 4 ζ (3) = 12 (2k + 3)32k k=0  
1 2j 2 √ X (−1) j−1 2ζ 2j, 3 − 3 − 1 ζ (2j) +2 3 (4 − 2j)! (2π )2j−1 j=1

(n ∈ N) ;

Evaluations and Series Representations

439

and "  #   ∞ X ζ (2k) 2 π2 1 log 3 + 2 + √ ζ 2, − 4ζ (2) ; ζ (3) = 13 3 (k + 1)32k π 3 k=0 " ∞ X (2π )2n ζ (2k) n−1
log 2 + 4 ζ (2n + 1) = (−1) (2k + 2n + 1)42k (2n + 1)! 22n − 1 k=0 ! n−1 j 2j X (−1) 2 −1 + (2n + 1)! ζ (2j + 1) (2n − 2j + 1)! (2π)2j j=1   
nX +1 ζ 2j, 41 −22j−1 22j − 1 ζ (2j) (−1) j  − (2n + 1)! (2n − 2j + 2)! (2π)2j−1

(n ∈ N)

j=1

and " ∞ X ζ (2k) (2π )2n
log 2 + 2 ζ (2n + 1) = (−1) (k + n)42k (2n)! 24n+1 + 22n − 1 k=0 ! n−1 X (−1) j 22j − 1 + (2n)! ζ (2j + 1) (2n − 2j)! (2π )2j j=1   
1 2j−1 22j − 1 ζ (2j) n j ζ 2j, X 4 −2 (−1)  − (2n)! (2n − 2j + 1)! (2π)2j−1 j=1  ∞ X 2π 2  ζ 2k) ζ (3) = log 2 + 4 9 (2k + 3)42k k=0   
1 2 X (−1) j−1 ζ 2j, 4 − 22j−1 22j − 1 ζ (2j)  +6 (4 − 2j)! (2π )2j−1 n−1

(n ∈ N);

j=1

and "    # ∞ X ζ (2k) 1 1 2π 2 log 2 + 2 + ζ 2, − 6ζ (2) ; ζ (3) = 35 4 (k + 1)42k π k=0 " ∞ X (2π )2n 4 ζ (2k) n−1

− ζ (2n + 1) = (−1) 2n 2n (2n + 1)! (2k + 2n + 1)62k 2 −1 3 −1 k=0 !

n−1 X 22j − 1 32j − 1 (−1) j + ζ (2j + 1) (2n − 2j + 1)! (2π )2j j=1     
n+1 ζ 2j, 13 + ζ 2j, 16 − 22j−1 32j −1 ζ (2j) 1 X (−1) j  +√ (2n−2j + 2)! (2π)2j−1 3 j=1

(n ∈ N)

440

Zeta and q-Zeta Functions and Associated Series and Integrals

and " ∞ 2 X ζ (2k) (2π )2n ζ (2n + 1) = (−1) 22n + 32n + 62n − 1 (2n)! (k + n)62k k=0

! n−1 X 22j − 1 32j − 1 (−1) j − ζ (2j + 1) (2n − 2j)! (2π )2j j=1     
n ζ 2j, 13 +ζ 2j, 16 − 22j−1 32j −1 ζ (2j) 1 X (−1) j  −√ (2π)2j−1 3 j=1 (2n−2j+1)! " ∞ X ζ (2k) π2 ζ (3) = − 2 18 (2k + 3)62k k=0     
1 1 2j−1 32j − 1 ζ (2j) 2 j−1 ζ 2j, + 2j, √ X 3 6 −2 (−1)  + 3 (4 − 2j)! (2π )2j−1 n−1

(n ∈ N);

j=1

and "∞      # π 2 X ζ (2k) 1 1 1 ζ (3) = + + ζ 2, − 16ζ (2) . √ ζ 2, 12 3 6 (k + 1)62k 2π 3 k=0 (Srivastava, Glasser and Adamchik [1097]) 15. Continuing Problems 13 and 14, show that ∞ X k=0

   2j Z∞ n ζ (2k) 1 X (−1) j−1 2n + 1 2 t2j+1 sech2 t dt, = 2k 2 2j + 1 2j π (2k + 2n + 1)2 j=0

0

for n ∈ N0 ; ∞ X

ζ (2k) (2k + 2n + 1)22k k=0
  n 2n + 1 (2j)! 22j − 1 1X 1 (−1) j−1 = ζ (2j + 1) − log 2, 2 2j 2 (2π )2j j=1

for n ∈ N0 ; " ∞ X (2π )2n ζ (2k)
log 2 + 2 ζ (2n + 1) = (−1) 2n (2k + 2n + 1)22k (2n + 1)! 2 − 1 k=0 
  n−1 2j − 1 X (2j)! 2 2n + 1 + (−1) j−1 ζ (2j + 1) , 2j (2π )2j n−1

j=1

for n ∈ N. (Srivastava, Glasser and Adamchik [1097, p. 842 et seq.])

Evaluations and Series Representations

441

16. Continuing Problems 13, 14 and 15, define Sp :=

∞ X k=0

π ζ (2k) =− 2k 2ω (2k + p)ω

Z1

tp cot



0

 πt dt ω

(p ∈ N; |ω| > 1) .

Show that Sp =

  Z πi 1 iω p (log z)p + − dz 2(p + 1)ω 2 2π 1−z

(p ∈ N; ω| > 1) ;

1

  Z1 πi 1 iω p dt {log (1 − (1 − )t)}p , Sp = − − 2(p + 1)ω 2 2π t 0

where, for convenience,   2π i  := exp (|ω| > 1) ; ω or, equivalently, ∞ X k=0

πi p! ζ (2k) = − 2(p + 1)ω 2 (2k + p)ω2k



iω 2π

p

  S1,p 1 − e2π i/ω

in terms of Nielsen’s generalized Polylogarithmic function Sn,p (z), defined by (cf., e.g., K¨olbig [686, p. 1233, Eq. (1.3)]) (−1)n+p−1 Sn,p (z) := (n − 1)!p! S1,p (z) = ζ (p + 1) +

Z1

0 p X k=0

(log t)n−1 {log (1 − zt)}p

dt t

(n, p ∈ N; z ∈ C) ;

(−1)k−1 {log(1 − z)}k Lip−k+1 (1 − z); k!

∞ X

  ζ (2k) πi 1 2π i/ω = − log 1 − e 2(p + 1)ω 2 (2k + p)ω2k k=0    p      1X p p! iω p iω k ζ (p + 1) + Lik e2π i/ω , − k! 2 2π 2 k 2π k=1

for p ∈ N and |ω| > 1; Z1

tp cot(νt)dt = −

0

+

  i 1 + log 1 − e2νi p+1 ν p! ν



i 2ν

p

ζ (p + 1) −

 p      1X p i k k! Lik+1 e2νi , ν k 2ν k=1

for p ∈ N and ν ∈ C \ {0}. (Srivastava, Glasser and Adamchik [1097, p. 843 et seq.])

442

Zeta and q-Zeta Functions and Associated Series and Integrals

17. Prove the following series representations for the values of L (s, χ ): (a) If χ (−1) = 1 and χ 6= 1, then     ∞ ∞ X 2lπx 2lπx χ (l) πx X χ (l) cos sin − p p p l2n+1 l2n l=1 l=1 "   n−1 2π x 2n X (−1)k−1 · k L (2k + 1, χ )

nL (2n + 1, χ ) − n = (−1)n

p

k=1

(2n − 2k) !

∞

+

τ (χ ) X (2k) ! L (2k, χ ) x2k p (2n + 2k) !

(2π x/p)2k # (n ∈ N) ;

k=1

(b) If χ (−1) = −1, then

L (2n, χ ) −

∞ X χ (l) l=1

l2n

 cos

2lπx p



2n−1 "X n−1

L (2k, χ ) (−1)k−1 (2n − 2k) ! (2πx/p)2k−1 k=1 # ∞ 2τ (χ ) i X (2k) ! 2k+1 + L (2k + 1, χ ) x (n ∈ N) , p (2n + 2k) !

= (−1)

n



2π x p

k=0

where L (s, χ ) is the Dirichlet L-function associated with a nontrivial Dirichlet character χ of modulus p and τ (χ ) denotes the Gauss sum defined by

τ (χ ) :=

p X k=1



2kπi χ (k) exp p

 

i :=

√

 −1 .

(Katsurada [638, p. 82, Theorem 3]) 18. Prove the following series representation: ∞ 1 X sin(2π lx) 2π x l2n+2 l=1 " 1 = (−1)n−1 (2π x)2n (2n + 1)!

ζ (2n + 1) +

+

n−1 X k=1

2n+1 X m=1

1 − log(2π x) m ∞

!

X (2k − 1)! ζ (2k) ζ (2k + 1) (−1) + 2 x2k (2n + 2k + 1)! (2n − 2k + 1)! (2π x)2k

#

k

k=1

(n ∈ N; x ∈ R; |x| ≤ 1). (Katsurada [638, p. 81, Theorem 2])

Evaluations and Series Representations

443

19. Prove the following series representations for the values of L (s, χ ): (a) If χ (−1) = 1 and χ 6= 1, then   ∞ 2lπx p X χ (l) sin 2π x p l2n+2 l=1  2n "X n−1 2π x L (2k + 1, χ ) (−1)k = (−1)n+1 p (2n − 2k + 1) ! (2π x/p)2k k=0 # ∞ X 2τ (χ ) (2k − 1) ! + L (2k, χ ) x2k (n ∈ N) ; p (2n + 2k + 1) !

L (2n + 1, χ ) −

k=1

(b) If χ (−1) = −1, then   ∞ p X χ (l) 2lπ x sin 2π x p l2n+1 l=1  2n−1 "X n−1 2π x L (2k, χ ) (−1)k = (−1)n+1 p (2n + 2k + 1) ! (2π x/p)2k−1 k=1 # ∞ X 2τ (χ ) i (2k) ! 2k+1 − L (2k + 1, χ ) x (n ∈ N) , p (2n + 2k + 1) !

L (2n, χ ) −

k=0

where L (s, χ ) and τ (χ ) are as those given in Problem 17. (Srivastava and Tsumura [1112]) 20. Prove the following series representations for ζ (3): 120 2 π 1573 ∞ X

ζ (3) = − ·

k=0

ζ (3) = −

8576k2 + 24286k + 17283 ζ (2k) ; (2k + 1) (2k + 2) (2k + 3) (2k + 4) (2k + 5) (2k + 6) (2k + 7) 22k

 6 2 1 π (80H4 − 31H5 − 49 log π) 17 120

+2

∞ X k=1

 32k + 49 ζ (2k) (n ∈ N) , (2k) (2k + 1) (2k + 2) (2k + 3) (2k + 4) (2k + 5) 22k

where Hk denotes the harmonic numbers, defined by 3.2(36). (Srivastava and Tsumura [1112, pp. 20 and 22])

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5 Determinants of the Laplacians During the last two decades, the problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention by many authors, including (among others) D’Hoker and Phong [377, 378], Sarnak [1004] and Voros [1201], who computed the determinants of the Laplacians on compact Riemann surfaces of constant curvature in terms of special values of the Selberg Zeta function. Although the first interest in the determinants of the Laplacians arose mainly for Riemann surfaces, it is also interesting and potentially useful to compute these determinants for classical Riemannian manifolds of higher dimensions, such as spheres. In this chapter, we are particularly concerned with the evaluation of the functional determinant for the n-dimensional sphere Sn with the standard metric. In computations of the determinants of the Laplacians on manifolds of constant curvature, an important roˆ le is played by the closed-form evaluations of the series involving the Zeta function given in Chapter 3 (cf., e.g., Choi and Srivastava [291, 292], Choi et al. [269]), as well as the theory of the multiple Gamma functions presented in Section 1.3 (cf., e.g., Voros [1201], Vardi [1190], Choi [260] and Quine and Choi [954]).

5.1 The n-Dimensional Problem Let {λn } be a sequence, such that 0 = λ0 < λ1 5 λ2 5 · · · 5 λn 5 · · · ;

λn ↑ ∞

(n → ∞);

(1)

hence, we consider only such nonnegative increasing sequences. Then, we can show that Z(s) :=

∞ X 1 , λsn

(2)

n=1

which is known to converge absolutely in a half-plane <(s) > σ for some σ ∈ R. Definition 5.1 (cf. Osgood et al. [881]). The determinant of the Laplacian 1 on the compact manifold M is defined to be Y det0 1 := λk , (3) λk 6=0 Zeta and q-Zeta Functions and Associated Series and Integrals. DOI: 10.1016/B978-0-12-385218-2.00005-0 c 2012 Elsevier Inc. All rights reserved.

446

Zeta and q-Zeta Functions and Associated Series and Integrals

where {λk } is the sequence of eigenvalues of the Laplacian 1 on M. The sequence {λk } is known to satisfy the condition as in (1), but the product in (3) is always divergent; so, for the expression (3) to make sense, some sort of regularization procedure must 0 be used. It is easily seen that, formally, e−Z (0) is the product of nonzero eigenvalues of 1. This product does not converge, but Z(s) can be continued analytically to a neighborhood of s = 0. Therefore, we can give a meaningful definition: det0 1 := e−Z (0) , 0

(4)

which is called the Functional Determinant of the Laplacian 1 on M. Definition 5.2 The order µ of the sequence {λk } is defined by ) ( ∞ X 1 <∞ . µ := inf α > 0 λαk

(5)

k=1

The analogous and shifted analogous Weierstrass canonical products E(λ) and E(λ, a) of the sequence {λk } are defined, respectively, by !) (  ∞  Y λ λ[µ] λ λ2 1− + ··· + (6) E(λ) := exp + [µ] λk λk 2λ2k [µ]λk k=1 and E(λ, a) :=

∞  Y 1− k=1

   λ λ λ[µ] , exp + ··· + λk + a λk + a [µ] (λk + a)[µ]

(7)

where [µ] denotes the greatest integer part in the order µ of the sequence {λk }. There exists the following relationship between E(λ) and E(λ, a) (see Voros [1201]):   [µ] m X λ E(λ − a) E(λ, a) = exp  Rm−1 (−a)  , (8) m! E(−a) m=1

where, for convenience, R[µ] (λ − a) :=

d[µ]+1 {− log E(λ, a)} . dλ[µ]+1

(9)

The shifted series Z(s, a) of Z(s) in (2) by a is given by Z(s, a) :=

∞ X k=1

1 . (λk + a)s

(10)

Determinants of the Laplacians

447

Formally, indeed, we have Z 0 (0, −λ) = −

∞ X

log(λk − λ),

k=1

which, if we define   D(λ) := exp −Z 0 (0, −λ) ,

(11)

immediately implies that D(λ) =

∞ Y

(λk − λ).

k=1

In fact, Voros [1201] gave the relationship between D(λ) and E(λ) as follows:   [µ] m X λ D(λ) = exp[−Z 0 (0)] exp − FPZ(m)  m m=1

 · exp −

[µ] X

C−m

m=2

m−1 X k=1

1 k

!

(12)

 λm  E(λ), m!

where an empty sum is interpreted to be nil and the finite part prescription is applied (as usual) as follows (cf. Voros [1201, p. 446]): ( f (s),  if s is not a pole, FPf (s) := (13) Residue , if s is a simple pole, lim f (s + ) −  →0

and Z(−m) = (−1)m m! C−m .

(14)

Now consider the sequence of eigenvalues on the standard Laplacian 1n on Sn . It is known from the work of Vardi [1190] (see also Terras [1148]) that the standard Laplacian 1n (n ∈ N) has eigenvalues µk := k(k + n − 1)

(15)

with multiplicity     k+n k+n−2 (2k + n − 1) (k + n − 2)! βkn := − = n n k! (n − 1)! n−2 2k + n − 1 Y = (k + j) (n − 1)! j=1

(16) (k ∈ N0 ) .

448

Zeta and q-Zeta Functions and Associated Series and Integrals


2 From now on, we consider the shifted sequence {λk } of {µk } in (15) by n−1 as a 2 fundamental sequence. Then, the sequence {λk } is written in the following simple and tractable form: 

n−1 λk = µk + 2

2

  n−1 2 = k+ 2

(17)

with the same multiplicity as in (16). We will exclude the zero mode, that is, start the sequence at k = 1 for later use. Furthermore, with a view to emphasizing n on Sn , we choose the notations Zn (s), Zn (s, a), En (λ), En (λ, a) and Dn (λ), whereas instead of Z(s), Z(s, a), E(λ), E(λ, a) and D(λ), respectively. We readily observe from (11) that  Dn

n−1 2

2 !

= det0 1n ,

(18)

where det0 1n denote the determinants of the Laplacians on Sn (n ∈ N). Choi [260] computed the determinants of the Laplacians on the n-dimensional unit sphere Sn (n = 1, 2, 3) by factorizing the analogous Weierstrass canonical product of a shifted sequence {λk } in (17) of eigenvalues of the Laplacians on Sn into multiple Gamma functions, whereas Choi and Srivastava [291, 292] and Choi et al. [269] made use of some closed-form evaluations of the series involving Zeta function given in Chapter 3 for the computation of the determinants of the Laplacians on Sn (n = 2, 3, 4, 5, 6, 7). Quine and Choi [954] made use of zeta regularized products to compute det0 1n and the determinant of the conformal Laplacian, det (1Sn + n(n − 2)/4). In following three sections, we compute the determinants of the Laplacians on Sn (n = 1, 2, 3, 4, 5, 6, 7) and det0 (1Sn + n(n − 2)/4), by using the aforementioned methods.

5.2 Computations Using the Simple and Multiple Gamma Functions Factorizations Into Simple and Multiple Gamma Functions We begin by expressing En (λ) (n = 1, 2, 3) as the simple and multiple Gamma functions. Our results are summarized in the following proposition (see Choi [260]; see also Voros [1201])

Determinants of the Laplacians

449

Proposition 5.1 The analogous Weierstrass canonical products En (λ) (n = 1, 2, 3) of the shifted sequence {λk } in 5.1(17) can be expressed in terms of the simple and multiple Gamma functions as follows: 1 √ √ 2 , 0(1 − λ)0(1 + λ) √
√
 4 02 ( 12 ) ec1 λ 0 21 − λ 0 12 + λ E2 (λ) = √
√
 √
√
2 , π 1 − 2 λ 1 + 2 λ 02 12 − λ 02 12 + λ E1 (λ) = 

(1)

(2)

and √ √ ec2 λ 02 (1 − λ)02 (1 + λ) E3 (λ) = , √ √ 1 − λ {03 (1 − λ)03 (1 + λ)}2

(3)

where, for convenience, c1 := 2(γ − 1 + 2 log 2)

and

c2 := log(2π) − 32 ,

and γ is the Euler-Mascheroni constant, defined by 1.1(3). Proof. Here we will prove only the representation (2). First of all, it follows from 5.1(15) and 5.1(16) with n = 2 that the eigenvalues of 12 on S2 have the sequence µk = k(k + 1) with multiplicity

2k + 1

(k ∈ N0 )

and the corresponding shifted sequence {λk } of {µk } in (4) by is   1 2 λk = k + 2

with multiplicity

(4) 1 4

2k + 1 (k ∈ N0 ).

[or 5.1(17) with n = 2]

(5)

It is easily seen from the definition 5.1(5) that both of the sequences (4) and (5) are of order µ = 1.  From 5.1(6) we find that the analogous Weierstrass canonical product of the shifted sequence {λk } in (5) is  ∞ Y  E2 (λ) = 1 −  k=1

 λ



2k+1

λ    2  exp   2  k + 12 k + 12

.

(6)

450

Zeta and q-Zeta Functions and Associated Series and Integrals

To express E2 (λ) in terms of simple and multiple Gamma functions, we consider the associated product E2+, defined by

E2+ (z) :=

∞ Y

   1−

k=1

z k+

 exp 

1 2

2k+1



!

z2

z k + 12

 +  2  1 2 k+ 2

.

(7)

Letting λ = −w2 in (6) and using the definition (7), we obtain E2 (λ) = E2+ (iw) E2+ (−iw).

(8)

It is seen from 1.4(3) that the G-function itself is related to the Barnes product EB (λ) given by EB (λ) : =

  k ∞  Y λ2 λ λ exp + 2 1− k k 2k

(9)

k=1

λ = (2π) 2

e

1 2



(1+γ )λ2 −λ



G(1 − λ).

Thus, E2+ (λ) can be further decomposed to involve the Barnes G-function, or rather the shifted Barnes product:



EB z, −

1 2

 =

∞ Y k=1

   1−

z k−

1 2

k



!

 exp 

z k − 21

z2

 +  2  . 1 2 k− 2

(10)

It readily follows from (7) and (10) that

E2+ (z) =

o2 n  EB z, − 12
(1 − 2z)2 exp 4z + 4z2    −1 !   ∞ Y  2 z z   z  · 1 − exp + .      1 1 2   k+ 2 k+ 2 2 k+ 1 k=1  2

(11)

Obviously, the sequence {λk } with λk = k is of order µ = 1. It follows from 1.3(2) and 5.1(6) that the analogous Weierstrass canonical product E(λ) of this sequence is  ∞  Y λ eγ λ λ E(λ) = 1− ek = , k 0(1 − λ) k=1

(12)

Determinants of the Laplacians

451

which, in view of 5.1(9), yields R0 (λ) = −

d {log E(λ)} = −ψ(1 − λ) − γ , dλ

(13)

where ψ is the Psi (or Digamma) function defined by 1.3(1). Now, the shift equation (8) applied to (12) with a = − 12 and µ = 1 yields the classical result:   ! ! 1 ∞ 0 Y z z (γ +2 log 2)z  2  , (14) 1− exp = e k + 12 k + 12 0 12 − z k=0 which, by virtue of 1.1(14), immediately becomes √ (γ +2 log 2)z    ∞  Y z 1 πe z 1  . 1− + exp + = k 2 k 2 (1 − 2z) e2z 0 21 − z k=1 If we apply 1.1(14), 1.3(53), 2.2(4), 2.3(2) and (15) to (11), we easily find that  n  o2 0 12 − z EB z, − 12 h   i. E2+ (z) = √ 2 π (1 − 2z) 22z exp (2 + γ )z + 2 + π4 z2

(15)

(16)

Similarly, the shift equation 5.1(8) applied to (9) with a = − 21 and µ = 2 yields       1 π2 2 EB z, − 12 = (2π) 2 z exp (log 2)z + log 2 + z 8   (17) h n oi G 12 − z   . · exp 12 (1 + γ )z2 + γ z G 21 A combination of (16) and (17) gives     2 h i (2π)z 0 12 − z  G 12 − z    E2+ (z) = √ exp −2z + (γ − 1 + 2 log 2) z2 .  π (1 − 2z)  G 1 2 (18) Finally, it follows from (8) and (18) that    n    o2 0 12 − iw 0 21 + iw G 12 − iw G 12 + iw E2 (λ) = n  o2 π G 12 (1 − 2iw)(1 + 2iw) h i · exp −2(γ − 1 + 2 log 2) w2 .

(19)

452

Zeta and q-Zeta Functions and Associated Series and Integrals

√ Since λ = −w2 , we have w = ±i λ.√We see that √ the second member of (19) remains the same with w replaced by either i λ or −i λ. In view of this observation and the relationship G(z) =

1 , 02 (z)

(19) is seen to be equivalent to (2). This completes the proof of the proposition (2). Propositions (1) and (3) can be proven similarly.

Evaluations of det0 1n (n = 1, 2, 3) By making use of Proposition 5.1, we can compute the determinants of the Laplacians on Sn (n = 1, 2, 3) explicitly. The results of these computations (as well as our computation of det0 14 ) are summarized by Proposition 5.2 The following evaluations hold true: det0 11 = 4π 2 , i h 1 det0 12 = exp 12 − 4 ζ 0 (−1) = e 6 A4 = 3.195311496 . . . ,   ζ (3) 0 det 13 = π exp = 3.338851215 . . . , 2π 2

(20) (21) (22)

and   1 35, 639, 301 13 0 2 0 det 14 = exp − ζ (−1) − ζ (−3) 3 3 3 217 · 5 · 7   13 2 183, 758, 875 1 , = · A 3 · C 3 · exp 3 217 · 33 · 7 0

(23)

where A is the Glaisher-Kinkelin’s constant, defined by 1.4(2), and C is the mathematical constant, defined by 1.4(70). Proof. det0 11 : It follows from 5.1(15) and 5.1(16) with n = 1 that the sequence of eigenvalues of 11 on S1 is µk = k2

with multiplicity

2 (k ∈ N0 ),

(24)

which corresponds to the shifted sequence {λk } in 5.1(17). It is obvious from the definition 5.1(5) that the sequence {µk } in (24) (and also its shifted sequence {λk }) is of order µ = 12 . Thus, its corresponding Zeta series Z1 (s) is given by Z1 (s) =

∞ X 2 = 2 ζ (2s), k2s k=1

(25)

Determinants of the Laplacians

453

where ζ (s) is the Riemann Zeta function, defined by 2.3(1). It is seen from 5.1(4) and 2.2(20) that det0 11 = exp[−4 ζ 0 (0)] = 4π 2 , which proves the evaluation (20).



Alternatively, even though it is not necessary to do so in the case of det0 11 , we can also prove (20), by making use of 5.1(12), 5.1(18) and (1). Indeed, since the shifted sequence considered above is of order µ = 12 , it is found from 5.1(12), 5.1(18) and 1.1(13) that     det0 11 = D1 (0) = exp −Z10 (0) lim E1 (λ) = exp −Z10 (0) , λ→0

where Z1 (s) is the same as in (25) and the evaluation (20) follows immediately. det0 12 : In this case, we saw that the shifted sequence {λk } in (5) is of order µ = 1. Therefore, we find from 5.1(12) and 5.1(18) that det0 12 = D2

  1 4

h i = exp −Z2 0 (0) − 14 FPZ2 (1) lim E2 (λ). λ→ 14

(26)

From (5) and 5.1(2), we have Z2 (s) =

∞ X k=1



  2k + 1 2s s 2s = 2 − 2 ζ (2s − 1) − 4 , 1 k+ 2

(27)

where ζ (s) is the Riemann Zeta function, defined by 2.3(1). Since Z2 (s) has a simple pole only at s = 1 with its residue 1 (see Section 2.3), it is seen from 5.1(13), 2.2(19) with a = 1 and (27) that   1 FPZ2 (1) = lim Z2 (1 + ) − →0     1 = −4 + lim 22+2 − 2 ζ (2 + 1) − →0      1 21+2 − 2 2+2 = −4 + lim 2 − 2 ζ (2 + 1) − + →0 2   1+2  2 −2 = −4 + 2γ + lim , →0  which, upon employing l’Hoˆ spital’s rule, immediately yields FPZ2 (1) = 2γ + 4 log 2 − 4.

(28)

454

Zeta and q-Zeta Functions and Associated Series and Integrals

It is easy to see from 2.3(10), 1.7(7) and 2.1(31) that Z2 0 (0) = −

13 1 13 log 2 − 2 ζ 0 (−1) = − − log 2 + 2 log A. 6 6 6

It follows from (2) and 1.1(12) (with z replaced by n  o4 02 12



1 2

(29)

+ z) that

1 lim E2 (λ) = exp (γ − 1 + 2 log 2) 1 4 2 λ→ 4    √ √  G 12 − λ G 12 − λ  √ , · lim  √  1 λ→ 41 2 − λ cosh π i λ



which, upon considering the following limit relationships: √  − λ 1  √ = lim π λ→ 14 cosh πi λ G



1 2

G and

√  − λ = 1, √ 1 2− λ



lim

λ→ 14

1 2

(30)

yields

lim E2 (λ) =

λ→ 14

n  o4 02 12 4π

 exp

 1 (γ − 1 + 2 log 2) . 2

(31)

Finally, if we substitute from (28), (29) and (31) into (26) and make use of 2.1(31), we are led at once to the evaluation (21). det0 13 : From 5.1(17) with n = 3, we find that the shifted sequence {λk } (by 1) of eigenvalues of the Laplacian 13 on S3 is λk = (k + 1)2

with multiplicity (k + 1)2

(k ∈ N0 ),

(32)

which is easily seen to be of order µ = 23 . Therefore, the analogous Weierstrass canonical product and its accompanying Zeta series for this sequence are ∞  Y E3 (λ) = 1− k=1

λ (k + 1)2



λ

 exp

(k + 1)2

(k+1)2 (33)

and Z3 (s) =

∞ X (k + 1)2 k=1

(k + 1)2s

= ζ (2s − 2) − 1.

(34)

Determinants of the Laplacians

455

It follows from 5.1(12) and 5.1(18) that   det0 13 = D3 (1) = exp −Z3 0 (0) − FPZ3 (1) lim E3 (λ). λ→1

(35)

It is also seen that Z3 (s) has a simple pole only at s = 32 with its residue 21 . Thus, we have from 5.1(13), 2.2(13) (with a = 1) and 2.3(22) (with n = 1) that FPZ3 (1) = Z3 (1) = ζ (0) − 1 = − 23

(36)

and Z3 0 (0) = 2 ζ 0 (−2) = −

ζ (3) . 2π2

(37)

If we substitute from (36) and (37) into (35), we obtain 3 ζ (3) + det 13 = exp 2 2π2 

0



lim E3 (λ).

(38)

λ→1

By taking the limit as λ → 1 on both sides of (3) and applying 1.4(6) and Theorem 1.6(a), we obtain   lim E3 (λ) = 2π exp − 32 A,

(39)

λ→1

where, for convenience,

A = lim

  √ n  √ o−2     02 1 − λ 03 1 − λ 

λ→1  

1−λ

.

(40)

 

It is readily seen from 2.1(27), 2.1(28), 2.1(24) and 2.2(17) that   A = exp ζ 0 (0) + 2 ζ 0 (−1) + ζ 0 (−2) · B · C , where, for convenience, B and C are defined by   λ    1  1 (2π) 2     B := lim √   λ→1  1 − λ 0 1 − λ  and n h √
√ √
io C := lim exp −ζ 0 −2, 1 − λ − 2 λ ζ 0 −1, 1 − λ . λ→1

(41)

456

Zeta and q-Zeta Functions and Associated Series and Integrals

First, to determine the limit value B , consider h  i 1 λ    (2π) 2  1 B = lim √ · √ oλ  , √ n  λ→1   1+ λ  1− λ 0 1− λ √ λ and using 1.1(9) and 1.1(13), immediately becomes (    1 2 ) π 1 1 π 2 1 z −2z 2 lim = lim . B= z→0 z {0(z)}1−2z+z2 z→0 2 2 0(z)

which, upon setting z = 1 −

We, thus, find from 1.1(2) that  π 12  π 12 2 B= lim zz −2z = . z→0 2 2

(42)

Next, to evaluate the limit value C , by setting n = 1 in 2.2(4), we have ζ (s, a) = ζ (s, 1 + a) + a−s ,

(43)

which, upon differentiating with respect to s, yields ζ 0 (s, a) = ζ 0 (s, 1 + a) − a−s log a.

(44)

Therefore, we see from (44) that    √  √   √  √  ζ 0 −1, 1 − λ = ζ 0 −1, 2 − λ − 1 − λ log 1 − λ and    √  √   √ 2 √  ζ 0 −2, 1 − λ = ζ 0 −2, 2 − λ − 1 − λ log 1 − λ , and so  √  lim ζ 0 −1, 1 − λ = ζ 0 (−1) and

λ→1

 √  lim ζ 0 −2, 1 − λ = ζ 0 (−2),

λ→1

which yield   C = exp −2 ζ 0 (−1) − ζ 0 (−2) .

(45)

Finally, by substituting from (42) and (45) into (41) and using 2.2(20), we find from (39) that   lim E3 (λ) = π exp − 23 , (46) λ→1

which, when substituted into (38), proves the evaluation (22).

Determinants of the Laplacians

457

A proof of the evaluation (23) will be presented in Section 5.3 in which each of the results (21), (22) and (23) is obtained by means of the summation formulas of Chapter 3 for series of Zeta functions (see also Problem 3 at the end of this chapter).

5.3 Computations Using Series of Zeta Functions In this section, we compute det0 1n (n = 2, 3, 4, 5, 6, 7) in a markedly different way from that detailed in Section 5.2 for n = 1, 2, 3. Our evaluations, here, are based largely on the summation formulas of Chapter 3 for series of Zeta functions. det0 12 : In view of 5.2(26), 5.2(28) and 5.2(29), it suffices to evaluate E2 ( 14 ). It follows from 5.2(6) that "   Y 2k+1  # ∞  1 1 1 1− E2 = exp , (1) 4 2k + 1 (2k + 1)2 k=1

which, upon taking logarithms and applying the Maclaurin series expansion of log(1 + x), yields !   ∞ ∞ X 1 X 1 1 =− −1 log E2 4 n (2k − 1)2n−1 n=2 n=2 (2) ∞ ∞ X ζ (2n − 1) − 1 X ζ (2n − 1) + . =− n n · 22n−1 n=2

n=2

If we apply 3.4(520) and 3.4(529) on the right-hand side of (2), we obtain   1 7 γ log E2 = −1 + − log 2 + 6 log A, 4 2 6

(3)

which, upon combining 5.2(26) with 5.2(8), 5.2(29) and 2.1(31), proves the evaluation 5.2(21). det0 13 : By virtue of 5.2(38), it suffices to compute E3 (1). Indeed, letting λ = 1 in 5.2(33) and taking logarithms of the resulting equation with the aid of the Maclaurin series expansion of log(1 + x), we get ! ∞ X ∞ ∞ X X 1 ζ (2n) − 1 log E3 (1) = − = − , (4) 2n n+1 (n + 1)k k=2

n=1

n=1

which, in view of 3.4(571), immediately yields 3 log E3 (1) = log π − . 2 Now, if we make use of (5) in 5.2(38), we are led to the evaluation 5.2(22).

(5)

458

Zeta and q-Zeta Functions and Associated Series and Integrals

det0 14 : Letting n = 4 in the shifted sequence 5.1(17) of eigenvalues of 14 on S4 , we obtain the following sequence: 2  λk = k + 23

with multiplicity

1 6 (k + 1)(k + 2)(2k + 3)

(k ∈ N0 ),

(6)

which obviously has the order µ = 2. Now, it follows from 5.1(12) and 5.1(18) that   9 det0 14 = D4 4     (7) 9 81 81 9 . = exp −Z40 (0) − FPZ4 (1) − FPZ4 (2) − C−2 E4 4 32 32 4 We can express Z4 (s) for the sequence (6) in terms of the Riemann Zeta function as follows: ∞

1 X (k + 1)(k + 2)(2k + 3) Z4 (s) =  2s 6 k=1 k + 32 ∞

∞

22s X (k + 1)(k + 2) 22s X k(k + 1) = 6 6 (2k + 3)2s−1 (2k + 1)2s−1 k=1 k=2 ! ∞ ∞ X 22s X 1 1 = − 24 (2k + 1)2s−3 (2k + 1)2s−1 =

=

22s 24

k=2

k=2

∞ X

1

∞ X

1

k=1

(2k − 1)2s−3

k=1

(2k − 1)2s−1

−

−

(8)

1 32s−3

+

1 32s−1

! ,

which, in view of 2.3(1), becomes Z4 (s) =

   1  2s−3 1 2s−3 1 2 − 1 ζ (2s − 3) − 2 − ζ (2s − 1) 3 3 4     1 2 2s−3 1 2 2s + . − 3 3 8 3

(9)

We observe from Section 2.3 that Z4 (s) has simple poles at s = 1 and s = 2 with 1 and 16 , respectively. their residues − 24 Using 2.3(10) and 5.1(14), we obtain 1 9, 801, 047 C−2 = Z4 (−2) = − 12 3 2 2 ·3 ·5·7

(10)

 2869  1 7 Z40 (0) = log 2− 1440 · 32 + ζ 0 (−1) − ζ 0 (−3). 12 12

(11)

and

Determinants of the Laplacians

459

Now, we evaluate FPZ4 (1) and FPZ4 (2). Since Z4 (s) has simple poles at s = 1 and s = 2, we have to use the second case of the definition of FPf (s) in 5.1(13) to compute the finite parts of Z4 (s) for s = 1 and s = 2. Using the expression in (9) for Z4 (s) and 2.3(6) (or 2.2[19] with a = 1), we easily see that   1 FPZ4 (1) = lim Z4 (1 + ) + →0 24     1 1 22 − 1 31 1 2−1 = − − lim 2 − ζ (2 + 1) − + 72 3 →0 4 2 4 γ 1 31 − log 2. =− − 72 12 6

(12)

Similarly, we have



 1 FPZ4 (2) = lim Z4 (2 + ) − →0 6 h  16 7 1 = − − ζ (3) + lim 22+1 − 1 81 12 3 →0    1 22+1 − 2 + · ζ (2 + 1) − 2 2 2 7 16 γ = − + + log 2 − ζ (3). 81 3 3 12

(13)

Since the sequence in (6) is of order µ = 2, its analogous Weierstrass canonical product E4 (λ) is

∞ Y



 1 (k+1)(k+2)(2k+3)

λ   E4 (λ) = 1 −  2  3 k=1 k+ 2 

6





(14)

λ2

λ 1   · exp  (k + 1)(k + 2)(2k + 3)   2 +  4  . 6 k + 32 2 k + 32

Upon setting λ = 49 in (14) and taking the logarithms on both sides of the resulting equation, if we make use of 2.3(1) and the Maclaurin series expansion of log(1 + x),

460

Zeta and q-Zeta Functions and Associated Series and Integrals

we obtain (∞ )   ∞ X 1X 9 32n =− (k + 1)(k + 2) log E4 4 6 n(2k + 3)2n−1 n=3 k=1 " # ∞ ∞ ∞ X 1 X 32n X 1 1 1 1 =− − − + 24 n (2k − 1)2n − 3 (2k − 1)2n−1 32n−3 32n−1 n=3 k=1 k=1 "  2n ∞ 1 X 1 2n 3 =− 3 ζ (2n − 3) − 8 ζ (2n − 3) − 32n ζ (2n − 1) 24 n 2 n=3 #  2n 3 +2 ζ (2n − 1) − 24 . (15) 2 Now, let α1 , α2 , α3 , and α4 denote the sums of the Zeta series occurring in 3.4(726), 3.4(729), 3.4(732) and 3.4(736), respectively. We then find from (15) that   1 9 = − (2α1 − α2 + α3 − 8α4 ) log E4 4 24 4 21 189 =− + γ − ζ (3) 9 32 128  979  17 5 + log A + log C + log 2− 1440 · 3 . 4 4

(16)

Finally, in view of 2.1(31) and 2.3(27), we are easily led from (7) and (9) through (16) to the evaluation 5.2(23). det0 15 : To evaluate det0 15 , we begin by setting a = 3, n = 1, n = 2 and t = 2 in 3.2(64), and then use 2.2(4), 2.2(20) and 2.3(22). We, thus, obtain ∞   X ζ (2n, 3) 2n+2 2 = 10 + log 3 · π −4 n+1

(17)

∞   X ζ (2n, 3) 2n+4 13ζ (3) 288 −16 2 = 20 − + log 2 · 3 · π . n+2 π2

(18)

n=1

and

n=1

By setting n = 5 in 5.1(17), we find that the shifted sequence of eigenvalues of 15 of S5 is given as follows: (k + 2)2

with multiplicity

1 (k + 1)(k + 2)2 (k + 3) 12

(k ∈ N).

(19)

Determinants of the Laplacians

461

It is seen that the sequence in (19) has the order µ = 25 . We also have Z5 (s) =

∞ 1 X (k + 1) (k + 2)2 (k + 3) 12 (k + 2)2s k=1

1 1 = [ζ (2s − 4) − ζ (2s − 2)] + 12 3

(20) 

 1 1 . − 22s 22s−2

It is observed that Z5 (s) has simple poles at s = FPZ5 (1) = Z5 (1) = −

5 24

and

3 2

and s = 52 . We, therefore, have

FPZ5 (2) = Z5 (2) = −

1 5 ζ (2) − . 12 48

We also have C−2 =

1 Z5 (−2) = −8 2

and

Z50 (0) =

ζ (5) ζ (3) + + 2 log 2. 8π 4 24π 2

We, thus, find that ∞  Y E5 (λ) = 1− k=1

 · exp

λ (k + 2)2

1

12

(k+1)(k+2)2 (k+3)

  λ2 1 λ , (k + 1)(k + 2)2 (k + 3) + 12 (k + 2)2 2(k + 2)4

which, upon setting λ = 4 and taking logarithms on each side of the resulting equation and using (17) and (18), yields   ∞ 1 X 22n 1 1 log E5 (4) = − ζ (2n − 4) − ζ (2n − 2) + 2n−2 − 2n−4 12 n 2 2 n=3 # "∞ ∞ X 22n+2 1 X 22n+4 =− ζ (2n, 3) − ζ (2n, 3) 12 n+2 n+1 n=1

n=2

  π 2 13 ζ (3) −24 =− + + log 2 · π . 9 12 π 2 If we set n = 5 in 5.1(18) and use 5.1(12), we finally have det0 15 = D5 (4) = exp[−Z5 0 (0) − 4 FPZ5 (1) − 8 FPZ5 (2) − 8 C−2 ] E5 (4)   π 197 ζ (3) ζ (5) = 26 exp − − . 3 2 24 π 2 8 π 4

(21)

det0 16 : Next, by setting n = 6 in 5.1(17), we obtain the shifted sequence of eigenvalues of 16 on S6 as follows:   5 2 1 k+ with multiplicity (2k + 5)(k + 4)(k + 3)(k + 2)(k + 1). (22) 2 120

462

Zeta and q-Zeta Functions and Associated Series and Integrals

We see that ∞ 1 X (2k + 5)(k + 4)(k + 3)(k + 2)(k + 1) 120 (k + 25 )2s k=1    1 h 2s = 2 − 32 ζ (2s − 5) − 10 22s − 8 ζ (2s − 3) 1920   i + 9 22s − 2 ζ (2s − 1) − ( 25 )2s .

Z6 (s) =

(23)

1 3 , − 48 It is observed that Z6 (s) has simple poles at s = 1, 2 and 3 with its residues 640 1 and 120 , respectively. It is also seen that the sequence in (22) has the order µ = 3. Now, we can find that

31 0 7 3 0 log 2 + 2 log 5 − ζ (−5) + ζ 0 (−3) − ζ (−1), 960 96 320   3 FPZ6 (1) = lim Z6 (1 + ) − →0 640 Z60 (0) = −

484051

28 · 33 · 5 · 7

3 9323 3 γ+ log 2, + 160 28 · 32 · 52 320 4483 1 21 FPZ6 (2) = − 5 2 4 + ζ (3) − (γ + 2 log 2), 320 24 2 ·3 ·5 6 2 1 7 93 γ FPZ6 (3) = − 6 + + log 2 − ζ (3) + ζ (5), 60 30 24 320 5 1 2217581021 , C−2 = Z6 (−2) = − 15 2 2 2 · 3 · 5 · 7 · 11 =−

and 56 1 62451523 + 7 . C−3 = − Z6 (−3) = 18 4 2 6 2 · 3 · 5 · 7 · 11 · 13 2 · 3 We also see that   ∞
5
9
1 X λn 7 7 7 log E6 (λ) = − ζ 2n − 5, 2 − 2 ζ 2n − 3, 2 + ζ 2n − 1, 2 , 60 n 16 n=4

(24) which, for λ =

25 4 ,

yields "∞       ∞ 25 1 X ζ (2n + 1, 72 ) 5 2n+6 5 X ζ (2n + 1, 72 ) 5 2n+4 log E6 =− − 4 60 n+3 2 2 n+2 2 n=1 n=2 #   ∞ 9 X ζ (2n + 1, 72 ) 5 2n+2 + . (25) 16 n+1 2 n=3

Determinants of the Laplacians

463

By setting n = 0, 1, 2, a = 72 and t = 25 in 3.2(67) and using some identities recorded in previous and present chapters, we have   ∞ X ζ (2n + 1, 72 ) 5 2n+2 155 25 11 = − γ− log 2 n+1 2 12 4 12 n=1

−

5 5 log 3 − log 5 + 3 ζ 0 (−1); 2 2

(26)

  ∞ X ζ (2n + 1, 72 ) 5 2n+4 3325 625 3561553 = 6 − 5 γ+ 5 log 2 n+2 2 2 2 .5.3831 2 n=1

−

53 125 15 log 3 − log 5 + 6 ζ 0 (−1) + ζ 0 (−3); (27) 8 8 8

and   ∞ X ζ (2n + 1, 27 ) 5 2n+6 623005 15625 253849 485 = 8 2 − 6 log 3 γ+ 6 2 log 2 − n+3 2 32 2 ·3 2 ·3 2 ·3 ·7 n=1

238627 3133 0 3 log 5 + ζ (−1) + ζ 0 (−2) 32 8 2 15 0 63 0 507 0 ζ (−3) + ζ (−4) + ζ (−5). + 4 8 16

−

(28)

Applying (26), (27) and (28) to (25), we obtain   25 4639 1385 246717677 log E6 = − 10 3 + 8 2 γ − 8 2 log 2 4 2 ·3 2 ·3 2 · 3 · 5 · 7 · 3831 6053 0 1 0 651 118711 log 5 − 6 ζ (−1) − ζ (−2) − 6 ζ 0 (−3) + 6 40 2 ·3·5 2 ·3·5 2 ·5 21 413875 96875 1 − 5 ζ 0 (−4) − 6 ζ 0 (−5) − 11 2 ζ (3) + 12 ζ (5). 2 2 ·5 2 ·3 2 (29) If we set n = 6 in 5.1(18) and use 5.1(12), we get    25 25 0 det 16 = D6 = exp −Z6 0 (0) − FPZ6 (1) 4 4   625 15625 625 25 25
− FP Z6 (2) − FP Z6 (3) − C−2 + C−3 · E6 , 32 192 32 8 4 which, upon using the above computations, yields  116791 38441354615245651 1511 0 0 − 3990625 det 16 = 2 735552 · 5 960 · exp − − ζ (−1) 5441253801984 240  1 0 2023 0 1 0 11 0 − ζ (−2) − ζ (−3) − ζ (−4) − ζ (−5) . 40 960 32 480

(30)

464

Zeta and q-Zeta Functions and Associated Series and Integrals

det0 17 : By setting n = 7 in 5.1(17), we see that the shifted sequence of eigenvalues of 17 on S7 is given as follows: λk = µk + 9 = (k + 3)2 with multiplicity 1 (k + 1)(k + 2)(k + 3)2 (k + 4)(k + 5). 360

(31)

It is found that Z7 (s) =

∞ 1 X (k + 1)(k + 2)(k + 3)2 (k + 4)(k + 5) 360 (k + 3)2s

(32)

k=1

1 = [ζ (2s − 6) − 5 ζ (2s − 4) + 4 ζ (2s − 2)] − 3−2s , 360 1 which shows us that Z7 (s) has simple poles at s = 23 , 52 and 72 with their residues 180 , 1 1 − 144 and 720 , respectively. It is also observed that the sequence in (31) has the order µ = 27 . If we set n = 7 in 5.1(18) and use 5.1(12), we get  81 det0 17 = D7 (9) = exp −Z7 0 (0) − 9FPZ7 (1) − FP Z7 (2) 2   (33) 9 81 C−2 + C−3 · E7 (9). −243 FP Z7 (3) − 2 2

We can also easily verify each of the following evaluations: 243 7 , FP Z7 (1) = − , 2 60 π2 π4 161 FP Z7 (3) = − 4 6 − 4 3 + 2 4 2 , 2 ·3 ·5 2 ·3 2 ·3 ·5

C−2 = −

81 , 2

C−3 =

FP Z7 (2) = −

and Z7 0 (0) = −

ζ (7) ζ (5) ζ (3) − − + 2 log 3. 32 π 6 48 π 4 180 π 2

It is observed that ( ∞ Y E7 (λ) = 1− k=1

 1 (k+1)(k+2)(k+3)2 (k+4)(k+5) 360 λ 2 (k + 3)  1 · exp (k + 1)(k + 2)(k + 3)2 (k + 4)(k + 5) 360   ) λ λ2 λ3 · + + , (k + 3)2 2(k + 3)4 3 (k + 3)6

7 24 · 34

+

π2 , 540

Determinants of the Laplacians

465

which, upon setting λ = 9 and taking logarithms on each side of the resulting equation and considering the Taylor-Maclaurin expansion of log(1 − x), yields log E7 (9) = −

∞ ∞ 1 X ζ (2n, 4) 2n+4 1 X ζ (2n, 4) 2n+6 ·3 + ·3 360 n+3 72 n+2 n=1

n=1

∞ 1 X ζ (2n, 4) 2n+2 39 2 3 4 259 − ·3 − π + π + . 90 n+1 80 100 240

(34)

n=1

By setting a = 4, n = 1, 2, 3 and t = 3 in 3.2(64) and using some identities already recorded in this presentation, we have ∞ X ζ (2n, 4) 2n+2 63 ·3 = − 5 log 2 + 9 log 3 + 4 log 5 − 9 log π, n+1 2 n=1 ∞ X n=1

(35)

ζ (2n, 4) 2n+4 1071 ·3 = − 29 log 2 + 243 log 3 + 16 log 5 n+2 4 − 81 log π −

27 ζ (3) , π2

(36)

and ∞ X ζ (2n, 4) 2n+6 32211 ·3 = − 125 log 2 + 3645 log 3 − 503936 log 5 n+3 10 n=1

1215 ζ (3) 405 ζ (5) − 729 log π − + . 2π2 2π4

(37)

If we apply (35), (36) and (37) to (34), we get 137 789 − log 3 + 1400 log 5 5 2 ·5 20 21 ζ (3) 9 ζ (5) 39 2 3 4 + log π + π + π . − − 2 4 80 100 16 π 16 π

log E7 (9) = −

(38)

Finally, from (34) and (38), and other previously recorded results, we obtain   1230367 949 ζ (3) 13 ζ (5) ζ (7) 0 − 177 1400 20 det 17 = 3 ·5 · π · exp − + − + . 60 720 π 2 24 π 4 32 π 6 (39)

5.4 Computations using Zeta Regularized Products Quine and Choi [954] made use of the zeta regularized products to compute det0 1n and the determinant of the conformal Laplacian, det (1Sn + n(n − 2)/4), which is

466

Zeta and q-Zeta Functions and Associated Series and Integrals

introduced here. In general, the conformal Laplacian is defined to be 1+

(n − 2) K , 4(n − 1)

where 1 is the Laplacian and K is the scalar curvature. For the sphere Sn , K = n(n − 1). Computation of the above determinants is equivalent to computing derivatives at s = 0 of the zeta function ∞ X k=1

βkn [k(k + n − 1)]s

for the Laplacian and ∞ X k=0

βkn [(k + n/2)(k + n/2 − 1)]s

for the conformal Laplacian and βkn being given in 5.1(16). For simplicity, we restrict our discussion of the conformal Laplacian to the case when n is even. We consider the more general zeta function as in Section 5.1, Zn (s, a) =

∞ X k=1

βkn [(k + a)(k + n − 1 − a)]s

for integers a, 0 5 a 5 n − 1, with a = 0 corresponding to the Laplacian and a = n/2 to the conformal Laplacian. If a(n − 1 − a) 6= 0, then
det (1Sn + a(n − 1 − a)) a (n − 1 − a) exp −Zn0 (0, a) , and, if a(n − 1 − a) = 0, then
det0 1n = exp −Zn0 (0, a) . We show below (Theorem 5.4) that for integers a, det(1Sn + a(n − 1 − a)), if a(n − 1 − a) 6= 0 and (n − 1) det0 1n are both of the form exp τn (a) +

n X

! τkn (a) ζ 0 (−k + 1)

,

(1)

k=1

where ζ is the Riemann Zeta function defined by 2.3(1) and the numbers τn (a) and τkn (a) are rational numbers for which we give explicit expressions in terms of coefficients of the Taylor expansion of βkn about k = −a and k = −(n − 1)/2. Using (1) and the functional equation for the Riemann Zeta function 2.3(11) or 2.3(12), it is easy

Determinants of the Laplacians

467

to compute numerical values for the constants involved. Our computations show that τn (a) = 0 for n odd and τkn (a) = 0, if n and k have opposite parity. The technique that we introduce here for dealing with this computation, which is different from other approaches, is a lemma on zeta regularized products (Lemma 5.3), which may be useful in simplifying and understanding other computations of this type. This gives us, in addition, a way to factor our functional determinant det(1Sn + a(n − 1 − a)) into multiple Gamma functions, as shown in Section 5.2, generalizing an equation of Voros [1201] and giving an alternate approach to computing it for integral a, 0 5 a 5 n − 1 (Section 5.2). We remark that choosing a to be an integer makes things considerably simpler because βkn = 0 for k = −1, . . . , −(n − 1) and the computation reduces to computation of derivatives of the Riemann Zeta function. The same techniques could be used for noninteger values of a but would involve a more complicated expression involving derivatives of the Hurwitz zeta function.

A Lemma on Zeta Regularized Products and a Main Theorem Let {λk } be a sequence of nonzero complex numbers. If Z(s) in 5.1(2) converges for <(s) > s0 and has meromorphic continuation to a function meromorphic in <(s) > σ , for some σ < 0, with at most simple poles, then we say that the sequence is zeta regularizable. We define λ−s k = exp(−s log λk ), and the definition of Z depends on the choice of arg λk . For a zeta regularized sequence, we define the zeta regularized product (see 5.1(4)) Y

z λk


= exp −Z 0 (0) .

We also define the product (see 5.1(11)) Y

z


(λk − λ) = exp −Z 0 (0, −λ) ,

where, for λ 6= λk , we adopt the convention that arg (λk − λ) ∼ = arg λk for |λk | large. Q For λk , the sequence of nonzero eigenvaluesQof the Laplacian on a manifold, z λk is called the determinant of the Laplacian and z (λk + λ) the functional determinant, det0 (1 + λ), as in Section 5.1. Information on the formal properties of zeta regularized products can be found in [955] and [1201]. The use of zeta regularized products can be traced back as far as Barnes [97]. Our approach is to try to product Q factor a zeta regularized Q Q into simpler ones. One sees that the equality of z (λ2k − λ2 ) and z (λk − λ) z (λk + λ) is untrue without the introduction of an exponential factor. This factor can be computed from the relationship between the zeta regularized product and the Weierstrass product found in the above-cited references. We give a confirmation independent of this in the proof below. Lemma 5.3 Let λj be a zeta regularizable sequence, and let h be an integer, such that P∞ −h−1 < ∞. For <(s) sufficiently large, let Z(s) be as in 5.1(2) and for λ 6= λ , |λ j j=1 j |

468

Zeta and q-Zeta Functions and Associated Series and Integrals

let F(s) =

 ∞  −s X
−s
−s λ2j − λ2 − λj − λ − λj + λ .

(2)

j=1

Then, we have −F (0) = 0

[h/2] X j=1

 2j  1 λ 1 . Res Z(s) 1 + + · · · + s=2j 3 2j − 1 j

(3)

In terms of zeta regularized products, the result can be stated as  Y Y 
Y
0 2 2 = eF (0) z λj − λ z λj + λ , z λj − λ where F 0 (0) is given by (3). Proof. Define Gk (s) by subtracting terms of the binomial expansion of the summand of (2),  −s Gk (s) = λ2k − λ2 − (λk − λ)−s − (λk + λ)−s −

[h/2] X 

    −s −2s−2j −s −s−2j 2j λk −2 λk λ . j 2j

j=0

(4)

Elementary estimates show that G(s) =

∞ X

Gk (s)

k=1

converges for <(s) = 0. Since G0k (0) = 0, term by term differentiation gives G0 (0) = 0.

(5)

Now, summing (4) over the index k gives

F(s) =

[h/2] X  j=0

    −s −s Z(2s + 2j) − 2 Z(s + 2j) + G(s). j 2j

Differentiating (6) at s = 0 and using (5) gives the result.

(6)



Determinants of the Laplacians

469

Theorem 5.4 For n ∈ N \ {1} and a an integer, such that 0 5 a 5 n − 1 and <(s) > n, let Zn (s, a) =

∞ X

βkn [(k + a)(k + n − 1 − a)]−s ,

(7)

k=1

where βkn is given as in 5.1(16). For k ∈ N (1 5 k 5 n), define functions tjn (a) by the expansion n βk−a =

n X

tjn (a) kj−1 .

(8)

j=1

The derivative at zero of (7) is given by −Zn0 (0, a) = τn (a) −

n X 

1 + (−1)n+j tjn (a) ζ 0 (1 − j) − A,

(9)

j=1

where   [n/2]  X 1 n n−1 t2j τn (a) = 1 + (−1)n 2j 2 j=1 2j   1 n−1 1 −a · 1 + + ··· + 3 2j − 1 2

(10)

and A = log(n − 1), if a(n − 1 − a) = 0 and A = log{a(n − 1 − a)} otherwise. Proof. Let F(s) =

∞ X

 βkn {(k + a)(k + n − 1 − a)}−s

(11)

k=1 −s

−(k + a)

− (k + n − 1 − a)

−s



.

Substituting λ = (n − 1)/2 − a, h = n, gives us the situation described in Lemma 5.3. Hence, if for <(s) > n, we set e Zn (s, λ) =

∞ X k=1

 −s n−1 βkn k + +λ , 2

(12)

we have F (0) = 0

[n/2] X j=1

  2j 1 1 λ e Res Zn (s, 0) 1 + + · · · + . s=2j 3 2j − 1 j

(13)

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Zeta and q-Zeta Functions and Associated Series and Integrals

We, now, show that Res e Zn (s, 0) = tkn



s=k

n−1 2

 (14)

for 1 5 k 5 n − 1 (k ∈ N). Using (9), we may write    ∞ X n X n − 1 −s+j−1 n−1 e `+ Zn (s, 0) = tjn 2 2 `=1 j=1

=

n X

tjn

j=1



n−1 2

   n+1 ζ s − j + 1, , 2

where ζ (s, a) is the Hurwitz Zeta function, defined by 2.2(1). Now, (14) easily follows.  Now, we find from (11) that Zn0 (0, a) = F 0 (0) + e Zn 0 (0, λ) + e Zn 0 (0, −λ),

(15)

and we need to compute the last two terms on the right side of (15). First, note that βkn = 0 for k = −1, −2, · · · , −(n − 1) and β0n = 1. Hence, if a 6= 0, we have 0

e Zn (0, −λ) =

∞ X

βkn (k + a)−s

k=1

=

∞ X

n βk−a k−s − a−s

k=1

=

∞ X

tkn (a) ζ (s − k + 1) − a−s .

k=1

Thus, we have 0

e Zn (0, −λ) =

n X

tkn (a) ζ 0 (−k + 1) + log a (a 6= 0).

(16)

k=1

If a = 0, we have   X n n−1 0 e Zn 0, − = tkn (0) ζ 0 (−k + 1). 2

(17)

k=1

Similarly, 0

e Zn (0, λ) =

n X k=1

tkn (n − 1 − a) ζ 0 (−k + 1) + log(n − 1 − a)

(a 6= n − 1),

(18)

Determinants of the Laplacians

471

and, for a = n − 1,  X  n n−1 0 e = tkn (0) ζ 0 (−k − 1). Zn 0, 2

(19)

k=1

n From βkn = (−1)n+1 β−k+n+1 , we deduce

tkn (n − 1 − a) = (−1)n+k tkn (a) .

(20)

Now, combining (13)–(20), we get (11).

Computations for small n The following is a table of det0 1n , for n = 2, . . . , 6, and the determinants of the conformal Laplacians det(1Sn + n(n − 2)/4), for n = 4, 6, 8. The first expression in each set of equations is obtained from Theorem 5.4. The second expression is obtained by replacing the values of ζ 0 at negative integers by its equivalent expression in terms of values of ζ and ζ 0 at positive integers obtained by differentiating the functional equation 2.3(14) or 2.3(22). Numeric evaluation of ζ 0 (k) (k ∈ N \ {1}) is discussed in [1191]. The computations below were done using Mathematica. det0 12 = exp = exp



 1 − 4 ζ 0 (−1) 2



2 ζ 0 (2) 1 1 + (γ + log(2π)) − 6 3 π2



∼ = 3.19531.
1 exp −2 ζ 0 (−2) − 2 ζ 0 (0) 2   ζ (3) 1 = exp log(2π) + 2 2π2

det0 13 =

∼ = 3.33885.   15 2 ζ 0 (−3) 13 ζ 0 (−1) 1 0 − − det 14 = exp 3 16 3 3   1 1267 16 13 ζ 0 (2) ζ 0 (4) = exp + (γ + log(2π)) − + 3 2160 45 6π2 2π4 ∼ = 1.73694.

472

Zeta and q-Zeta Functions and Associated Series and Integrals

 0  1 ζ (−4) 23 ζ 0 (−2) exp − − − 2 ζ 0 (0) 4 6 6   1 23 ζ (3) ζ (5) − = exp log(2π) + 4 24 π 2 8π4 ∼ = 1.76292.   1 149 ζ 0 (−1) 455 ζ 0 (−5) det0 16 = exp − − 2 ζ 0 (−3) − 5 432 30 30  1 751 303733 = exp + (γ + log(2π)) 5 453600 1890  149 ζ 0 (2) 3 ζ 0 (4) ζ 0 (6) + − − 60 π 2 2π4 8π6 ∼ = 1.29002.  
1 2 ζ 0 (−3) ζ 0 (−1) det 1S4 + 2 = exp − − − 144 3 3   53 1 ζ 0 (2) ζ 0 (4) + = exp − + (γ + log(2π)) − 2160 45 6π2 2π4 ∼ = 1.04562.  
ζ 0 (−5) ζ 0 (−1) 1 − + det 1S6 + 6 = exp 1350 30 30   1 ζ 0 (2) ζ 0 (6) 1459 − (γ + log(2π)) + − = exp 453600 378 60 π 2 8π6 ∼ = 0.995257.  
ζ 0 (−7) ζ 0 (−5) ζ 0 (−3) ζ 0 (−1) 5497 − + + − det 1S8 + 12 = exp − 50803200 1260 360 360 210  391481 23 = exp − + (γ + log(2π)) 76204800 56700  ζ 0 (4) ζ 0 (6) ζ 0 (8) ζ 0 (2) − − + + 420 π 2 480 π 4 96 π 6 32 π 8 ∼ = 1.00069. det0 15 =

It should be noted that we need to compare the evaluation of det0 14 in Sections 5.3 and 5.4 with those in other works (see Section 5.5).

5.5 Remarks and Observations Even though each of the determinant expressions 5.2(20), 5.2(21) and 5.2(22) has been considered by other workers on this subject, our novel approach in Section 5.3 to the familiar problem of computation of the determinants of the Laplacians on the

Determinants of the Laplacians

473

n-sphere Sn (n = 1, 2, 3) is relatively more natural than the earlier approaches to these and analogous problems. The evaluations of det0 1n (n = 1, 2, 3) in Choi [260], Quine and Choi [954], Choi and Srivastava [291] and Kumagai [706] have the same results, whereas the evaluation of det0 14 in Quine and Choi [954] is equal to that in Kumagai [706] and is not equal to that in Choi and Srivastava [292] (see also Section 5.3). It is the only difference that the number 35, 639, 301/(217 · 5 · 7), appearing in the evaluation of det0 14 in Choi and Srivastava [292] (see also Proposition 5.2), was evaluated as 15/16 in Quine and Choi [954, p. 726] and Kumagai [706, p. 201, Corollary (iv)]. It seems to need to examine C−m in 5.1(14) (see Voros [1201, p. 444, Eq. (3.3)]) again. Vardi [1190] also computed det0 1n , by employing a direct analysis of the involved Zeta series in 5.1(2) with the original (or unshifted) sequence {µk } in 5.1(15). The results of det0 1n (n = 1, 2) in Proposition 5.2 agree with those of Vardi’s (1988b). Yet Vardi’s ingenious computation for det0 1n should have some gaps (see Problem 5 below). Most recently, Kumagai (1999) evaluated det0 1n , by filling the gaps in Vardi’s work (1988b), including a rather detailed discussion for the computations of the det0 1n worked by several earlier authors (cf. Choi [260], Efrat [400], Quine and Choi [954], Vardi [1190]). Quine and Choi [954] computed det0 1n and det(1n + n(n − 2)/4) the determinants of the Laplacians and the conformal Laplacians (respectively) on sphere, by using techniques due to Weisberger [1219] (see Section 5.4). Yet, their technique has the advantage of offering a unified approach. Branson and Ørsted [171] computed the conformal determinant det(14 + 2), which agrees with the first expression as in Section 5.4 above (see also Quine and Choi [954, p. 727]). Quine et al. [955] introduced the Zeta regularized product for the general sequences of nonzero complex numbers, instead of the increasing sequences of nonnegative real numbers given in 5.1(2), in which case their Zeta regularized product corresponds to the determinant of the Laplacian, defined by 5.1(4). They gave many examples for this extended concept of the determinant of the Laplacian. Osgood et al. [881] noted that the modern quantum geometry of strings is concerned (to a large extent) with the set of all surfaces, varying metrics on those surfaces and determinants of the associated Laplacians. They investigated this determinant as a function of the metric on a given surface and, in particular, its extreme values when the metric is suitably restricted. They also observed an interesting fact related to the study of this chapter: For all metrics σ on S2 of area 4π, i h (1) det0 1σ 5 exp 21 − 4 ζ 0 (−1) = det0 12 , where the equality holds true, if and only if σ is the standard round metric, 1σ being the Laplacian on S2 with the given metric σ.

Problems P f (n) n−s does not converge for all s or is divergent for all s. 1. Suppose that the series Show that there exists a real number σa , called the abscissa of absolute convergence, such

474

Zeta and q-Zeta Functions and Associated Series and Integrals

P that the series |f (n) n−s | converges absolutely, if <(s) = σ > σa , but does not converge absolutely, if, σ < σa . (cf. Apostol [65, p. 225]) 2. Show that the analogous Weierstrass canonical product En (λ) of the shifted sequence {λk } in 5.1(17) for the standard Laplacian 1n on the n-dimensional unit sphere Sn can be expressed in terms of the multiple Gamma functions 0n given in Section 1.3 as follows:   √  √  n−1 0n−1 n−1 − λ 0 + λ n−1 2 2 En (λ) = Rn (λ) n    o2 , √ √ n−1 0n n−1 2 − λ 0n 2 + λ  2 where Rn (λ) is a meromorphic function with a simple pole only at λ = n−1 and 01 (z) = 2 0(z). Show also that the above formula for En (λ) holds true for all n ∈ N, if we define 00 (z) := 1z . (cf. Choi [259]; see also Theorem 1.4) 3. Let Rn (λ) be as in Problem 2 above. By using Proposition 5.1, show that R1 (λ) = − λ1 , n  o4 02 21 exp [2(γ − 1 + 2 log 2)λ] , R2 (λ) = π (1 − 4 λ) and R3 (λ) =

h  i 1 exp log(2π ) − 23 λ . 1−λ

Also determine R4 (λ) explicitly and evaluate det0 14 with the explicit form of E4 (λ), by using the method presented in Section 5.2. 4. Show why Vardi’s arguments leading to the evaluation:   det0 13 = exp −2 ζ 0 (−2) + ζ 0 (−1) − 3 ζ 0 (0) cannot be justified. (cf. Vardi [1190]) 5. Prove the inequality 5.5(1). (cf. Osgood et al. [881]) 6. Let M be a compact Riemann surface of genus g ≥ 2. M can be identified with H/ 0, the action of a fuchsian group 0 on the upper complex half-plane H = {z = x + iy | y > 0}
endowed with the Poincar´e metric ds2 = y−2 dx2 + dy2 . This is the classic model for hyperbolic geometry of constant negative curvature, R = −1. 0 is a discrete subgroup of PSL(2, R) = SL(2, R)/{±1}, the group of M¨obius transformations. From the GaussBonnet theorem, one infers R A = 2π χ = 4π(1 − g), where A denotes the area of M and χ its Euler characteristic, that is, A = 4π(g − 1). In the Poincar´e metric, the Laplacian on M (Laplace-Beltrami operator) is given by 1 = y2 (∂ 2 /∂x2 + ∂ 2 /∂y2 ), and one is interested in the eigenvalue problem −1u = λu. The spectrum of 1 on M is discrete and real, 0 = λ0 < λ1 5 λ2 5 · · · with # (eigenvalues λn with λn 5 λ)∼ (A/4π) λ asymptotically (Weyl’s law).

Determinants of the Laplacians

475


Since the elements τ ∈ ac db ∈ 0 are hyperbolic, that is, |Trτ | = |a + d| > 2, they are conjugate with 0 to a M¨obius transformation of the form z → N(τ ) z, 1 < N(τ ) < ∞, where N(τ ) is called the norm of τ . (τ1 , τ2 ∈ 0 are conjugate within 0, if there exists a τ3 ∈ 0, such that τ1 = τ3 τ2 τ3−1 .) The class of all elements in 0 that are conjugate to a given τ is called the conjugacy class of τ in 0 and denoted by {τ }. The number N(τ ) is, of course, the same within a conjugacy class and measure the “magnification.” N(τ ) has, however, another striking geometric interpretation, since there exists a unique relationship between the conjugacy classes in 0 and the homotopy classes of closed paths on the surface M. In each class, one defines a length l(τ ) by the length of the shortest closed path measured by means of the Poincar´e distance. One, then, obtains N(τ ) = exp[l(τ )], l(τ ) > 0. Thus, the conjugacy classes in 0 can be uniquely parameterized by their length spectrum {l(τ )}. Given any τ ∈ 0, there is a unique τ0 , such that τ = τ0n , n ∈ N; τ0 is called primitive element of 0, since it cannot be expressed as a power of any other element of 0. The corresponding closed orbit with length l(τ0 ) is called a prime geodesic on M. Obviously, l(τ ) = l(τ0n ) = n l(τ0 ), since in this case the prime geodesic is traversed n times. For the length spectrum of M, one has Huber’s law ν(x) ∼ ex /x (x → ∞), where ν(x), is the number of inconjugate primitive τ ’s with l(τ ) ≤ x. (a) Prove the Selberg trace formula: ∞ X n=0

A h(τn ) = 2π +

Z∞ t tanh(π t) h(t) dt −∞

∞ XX {τ }p n=1

l(τ )   g(n l(τ )), sinh 21 nl(τ )

where all series and the integral converges absolutely under the following conditions on the function h(t): (i) h(−t) = h(t), (ii) h(t) is analytic in a strip |= t| ≤ 12 +  ( > 0), (iii) |h(t)| ≤ a(1 + |t|2 )−1− (a > 0); the function g(u) is the Fourier transform of h(t): Z∞

1 g(u) = 2π

exp(−iut) h(t) dt; −∞

the sum on the left-hand side runs over the eigenvalues of 1 parameterized in the form λn = 14 + τn2 , that is, over the pairs (τn , −τn ), τn ∈ C (τ = 0 has to be counted twice, if 41 happens to be an eigenvalue); the sum on the right-hand side is taken over all primitive conjugacy class in 0, denoted by {τ }p . By choosing 1

h(t) = t2 +



s−

1 2

2 −

1 t2 +



z−

1 2

2

(<(s), <(z) > 1)

in the Selberg trace formula in (a), (b) Prove that  ∞  X 1 1 − = −2(g − 1)[ψ(s) − ψ(z)] λn + s(s − 1) λn + z(z − 1) n=0

+

1 Z 0 (s) 1 Z 0 (z) − , 2s − 1 Z(s) 2z − 1 Z(z)

476

Zeta and q-Zeta Functions and Associated Series and Integrals

where ψ is the Psi (or Digamma) function given in Section 1.2 and Z(s) denotes the Selberg Zeta function on M, defined by Z(s) :=

∞ YY

{1 − exp [−(s + n) l(τ )]} .

{τ }p n=0

(c) Prove that 1 1 Z 0 (s) =B+ + (g − 1) [ψ(s) + γ ] 2s − 1 Z(s) s(s − 1)   ∞ X 1 1 + − , λn + s(s − 1) λn n=1

where γ is the Euler-Mascheroni constant and   1 1 Z 00 (1) 1 Z 0 (z) − = . B := lim z→1+ 2z − 1 Z(z) z(z − 1) 2 Z 0 (1) (d) Deduce the following Laurent expansion about s = 1: ∞ X 1 Z 0 (s) 1 = + (B − 1) + an (s − 1)n , 2s − 1 Z(s) s−1 n=1 

an = (−1)n+1 1 + 2(g − 1) ζ (n + 1)

+

[n/2] X

(−1)j+1



j=0

  n−j ζ1 (n + 1 − j) j

where ζ (s) is the Riemann Zeta function and ζ1 (s) denotes ζ1 (s) :=

∞ X 1 . λsn n=1

(e) Prove that the finite part of ζ1 (s) at s = 1 is FP ζ1 (1) = 2(g − 1) γ + B (:= γ1 ) . (f) Prove that ζ10 (0) = 2(g − 1) C − log(Z 0 (1)), where π C := − 2

Z∞ 0

t2 +

1 4



  1 2 1 − log t + dt 4 cosh2 (π t)

1 1 = − log(2π ) − 2 ζ 0 (−1). 4 2

(n ∈ N),

Determinants of the Laplacians

477

Consider the functional determinant of 1 on M D1 (z) := det0 (−1 + z)

(z ∈ C),

where the prime indicates that the zero mode has been omitted and (by zeta regularization) D1 (0) := exp [−ζ10 (0)]. (g) Prove that ∞  Y

  z exp − λn n=1    ∞  Y z z = Z 0 (1) exp [γ1 z − 2(g − 1) C] 1+ exp − . λn λn

D1 (z) = D1 (0) exp (γ1 z)

1+

z λn



n=1

(h) Prove the following fundamental relation for the Selberg Zeta function: Z(s) = Z 0 (1) s(s − 1) V(s), where, for convenience, n iA 2π V(s) : = exp [γ1 s(s − 1)] (2π )1−s exp[s(s − 1)] G(s) G(s + 1)    ∞  Y s(s − 1) s(s − 1) exp − , · 1+ λn λn n=1

where G(s) denotes the Barnes G-function given in Section 1.3. (i) Prove that n i2(g−1) . Z(s) = s(s − 1) D1 (s(s − 1)) (2π )1−s exp[C + s(s − 1)] G(s) G(s + 1) (Cf. Steiner [1129]; see also Voros [1200]) 7. Consider the function z(s, a, b, c) =

∞ X n=1

Pd (cn) , (cn + a)s (cn + b)s

of the complex variable s for <(s) > d+1 2 , where Pd is a polynomial of degree d and a, b, c real constants with a, b > −1, c > 0. Introduce the function of the complex variable z: φ(z, s, a, b, c) =

∞ X n=1

Show that: For <(s) > z(s, a, b, c) =

Pd (z) . (cz + a)s (cz + b)s

d+1 2 ,

1 φ(1, s, a, b, c) + 2

Z∞

φ(z, s, a, b, c) dz + I(s),

1

where I(s) is an entire function of s. (See Spreafico [1063, Proposition 1])

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6 q-Extensions of Some Special Functions and Polynomials

The theory of hypergeometric functions in one, two and more variables leads to a natural unification of much of the material of concern to the mathematical analysts from the seventeenth century to the present day. Functions of this type may be generalized along the lines of basic (or q-) number, resulting in the formation of q-extensions (or q-analogues). This was first systematically effected by E. Heine (1821–1881) in the middle of the nineteenth century, and the work was subsequently greatly extended by F. H. Jackson (1870–1960), W. N. Bailey (1893–1961), L. J. Slater, G. E. Andrews and many others up to the present day. In fact, in recent years, various families of basic (or q-) series and basic (or q-) polynomials have been investigated rather widely and extensively due mainly to their having been found to be potentially useful in such wide variety of fields as (for example) theory of partitions, number theory, combinatorial analysis, finite vector spaces, Lie theory, particle physics, nonlinear electric circuit theory, mechanical engineering, theory of heat conduction, quantum mechanics, cosmology and statistics (see, for details, [1099, pp. 350–351]). The books and monographs by (among others) W. N. Bailey [87], L. J. Slater [1040], H. Exton [439], H. M. Srivastava and P. W. Karlsson [1099, Chapter 9] and G. Gasper and M. Rahman [468] discussed extensively basic (or q-) hypergeometric functions in one, two and more variables. Here, in the present chapter, we choose to investigate some remarkable qdevelopments around (especially) the Zeta and related functions, which are reported in recent years. We also present the background material, involving (for example) Jackson’s q-integral, the q-Gamma function and the q-Beta function, multiple q-Gamma functions, q-Bernoulli numbers and q-Bernoulli polynomials, q-Euler numbers and q-Euler polynomials, q-Zeta functions, multiple q-Zeta functions and so on.

6.1 q-Shifted Factorials and q-Binomial Coefficients The q-shifted factorial (a; q)n is defined by  1 Q (a; q)n := n−1  (1 − a qk )

(n = 0) (n ∈ N),

k=0

Zeta and q-Zeta Functions and Associated Series and Integrals. DOI: 10.1016/B978-0-12-385218-2.00006-2 c 2012 Elsevier Inc. All rights reserved.

(1)

480

Zeta and q-Zeta Functions and Associated Series and Integrals

where a, q ∈ C and it is assumed that a 6= q−m (m ∈ N0 ). It is noted that some other notations that have been used in the literature for the product (a; q)n in (1) are (a)q,n , [a]n , and even (a)n , when the Pochhammer symbol 1.1(5) is not used and the base q is understood. The q-shifted factorial for negative subscript is defined by 1 (1 − a q−1 ) (1 − a q−2 ) · · · (1 − a q−n )

(a; q)−n :=

(n ∈ N0 ),

(2)

which yields n (−q/a)n q(2) (a; q)−n = = (a q−n ; q)n (q/a; q)n

1

(n ∈ N0 ).

(3)

We also write ∞ Y

(a; q)∞ :=

(1 − a qk )

(a, q ∈ C; |q| < 1).

(4)

k=0

It is noted that, when a 6= 0 and |q| = 1, the infinite product in (4) diverges. So, whenever (a; q)∞ is involved in a given formula, the constraint |q| < 1 will be tacitly assumed. It follows from (1), (2) and (4) that (a; q)n =

(a; q)∞ (a qn ; q)∞

(n ∈ Z),

(5)

which can be extended to n = α ∈ C as follows: (a; q)α =

(a; q)∞ (a qα ; q)∞

(α ∈ C; |q| < 1),

(6)

where the principal value of qα is taken. A list of easily-verified useful identities is given below:  n q1−n ; q (−a)n q(2) (n ∈ Z); a n  −1 1−n   n n a q ; q = (a; q)n −a−1 q−(2) (n ∈ Z); a n  q   a n n (a q−n ; q)n = ;q − q−(2) (n ∈ Z); a q n

(a; q)n =



(a; q)n−k =

(a; q)n −1 (a q1−n ; q)k

(−q a−1 )k q(2)−n k k

(n, k ∈ Z);

(7) (8) (9) (10)

q-Extensions of Some Special Functions and Polynomials

(q−n ; q)k =

k (q; q)n (−1)k q(2)−n k (q; q)n−k

481

(n, k ∈ Z);

(a; q)k (q a−1 ; q)n −n k q (n, k ∈ Z); (a−1 q1−k ; q)n   a n−k (k)−(n) (q/a; q)n −n − q 2 2 (n, k ∈ Z); (a q ; q)n−k = (q/a; q)k q (a q−n ; q)k =

(a qk ; q)n−k =

(a; q)n (a; q)k

(n, k ∈ Z);

(11) (12) (13) (14)

(a; q)n+k = (a; q)n (a qn ; q)k

(n, k ∈ Z);

(15)

(a; q)k (a qk ; q)n (a; q)n

(n, k ∈ Z);

(16)

(a qn ; q)k =

(a q2k ; q)n−k =

(a; q)n (a qn ; q)k (a; q)2k

(a; q)2n = (a; q2 )n (a q; q2 )n

(n, k ∈ Z);

(n ∈ Z);

(18)

(a2 ; q2 )n = (a; q)n (−a; q)n (n ∈ Z);   a n −3 (n) (q/a; q)2n −2n − 2 q 2 (a q ; q)n = (q/a; q)n q (a q−kn ; q)n = (a qkn ; q)n =

(19) (n ∈ Z);

n (q/a; q)kn 2 (−a)n q(2)−k n (q/a; q)(k−1)n

(a; q)(k+1)n (a; q)kn

(a qj k ; q)n−k =

(17)

(n, k ∈ Z);

(n, k ∈ Z);

(a; q)n (a qn ; q)( j−1) k (a; q)j k

(20) (21) (22)

(n, k, j ∈ Z).

(23)

We, now, introduce some more q-notations, which would appear in this chapter quite frequently. The notation [z]q is defined by [z]q :=

qz − 1 1 − qz = 1−q q−1

(z ∈ C; q ∈ C \ {1}; qz 6= 1).

(24)

A special case of (24) when z ∈ N is [n]q =

qn − 1 = 1 + q + · · · + qn−1 q−1

(n ∈ N),

which is called the q-analogue (or q-extension) of n ∈ N, since lim [n]q = lim (1 + q + · · · + qn−1 ) = n.

q→1

q→1

(25)

482

Zeta and q-Zeta Functions and Associated Series and Integrals

The q-analogue of n! is then defined by  [n]q ! :=

1 [n]q [n − 1]q · · · [2]q [1]q

if n = 0, if n ∈ N,

from which the q-binomial coefficient (or the Gaussian polynomial analogous to is defined by   [n]q ! n := [n − k]q ! [k]q ! k q


n, k ∈ N0 ; 0 5 k 5 n .

(26) n
k )

(27)

It is easily seen from (1) and (27) that (q; q)n = (1 − q)n [n]q !

(n ∈ N0 ).

(28)

The q-binomial coefficient in (27) can be generalized in a similar way as in 1.1(20):   [α]q;k α := [k]q ! k q

(α ∈ C; k ∈ N0 ),

(29)

where [α]q;k is defined by [α]q;k := [α]q [α − 1]q · · · [α − k + 1]q

(α ∈ C; k ∈ N0 ).

(30)

The following notations are also frequently used: (a1 , a2 , . . . , am ; q)n := (a1 ; q)n (a2 ; q)n · · · (am ; q)n

(31)

(a1 , a2 , . . . , am ; q)∞ := (a1 ; q)∞ (a2 ; q)∞ · · · (am ; q)∞ .

(32)

and

Now, the generalized binomial coefficient in (29) can be further generalized as follows:
  qβ+1 , qα−β+1 ; q ∞ α
:= β q q, qα+1 ; q ∞

(α, β ∈ C; |q| < 1).

(33)

  It is noted that, whenever there is no confusion, the notations [z]q , [n]q ! and βα q   are simply written as [z], [n]! and βα , respectively. We record some known identities

q-Extensions of Some Special Functions and Polynomials

of [z]q and

α 

β q

[−n]q = − [n] 1 = q

as follows: 1 [n]q qn

(n ∈ Z);

1 [n]q qn−1

(n ∈ Z);

q

1 qn(n−1)/2

(34) (35)

(m, n ∈ Z);

(36)

[n]q ! (n ∈ Z);

(37)

[m n]q = [m]q [n]qm [n] 1 ! =

483

   
n n (q; q)n n, k ∈ N0 ; 0 5 k 5 n = = k q n − k q (q; q)k (q; q)n−k     [m]q m − 1 m = (m ∈ Z; n ∈ N); n q [n]q n − 1 q     1 m m = n(m−n) (m ∈ Z; n ∈ N0 ); n 1 n q q q         m k m m−j = (m ∈ Z; k, j ∈ N0 ; 0 5 j 5 k); k q j q j q k−j q
  q−α ; q k
k k α −qα q−(2) (α ∈ C; k ∈ N0 ) ; = (q; q)k k q
  qα+1 ; q k k+α = (α ∈ C; k ∈ N0 ) ; k q (q; q)k     k −α α+k−1 = (−q−α )k q−(2) (α ∈ C; k ∈ N0 ); k q k q       α+1 α α k = q + k q k q k−1 q     α α = + qα+1−k (α ∈ C; k ∈ N0 ) ; k q k−1 q n   X
k n (z; q)n = (−z)k q(2) z ∈ C; n, k ∈ N0 ; 0 5 k 5 n . k q

(38) (39) (40)

(41)

(42)

(43) (44)

(45)

(46)

k=0

6.2 q-Derivative, q-Antiderivative and Jackson q-Integral We begin by noting that F. J. Jackson was the first to develop q-calculus in a systematic way.

484

Zeta and q-Zeta Functions and Associated Series and Integrals

q-Derivative The q-derivative of a function f (t) is defined by Dq {f (t)} :=

dq f (qt) − f (t) {f (t)} = . dq t (q − 1)t

(1)

It is noted that lim Dq {f (t)} =

q→1

d {f (t)}, dt

if f (t) is differentiable. We record some easily derivable q-derivative formulas: Dq {a f (t) + b g(t)} = a Dq {f (t)} + b Dq {g(t)}; Dq {f (t) g(t)} = f (qt) Dq {g(t)} + g(t) Dq {f (t)},

(2) (3)

where the functions f (t) and g(t) are obviously interchangeable;  Dq

 g(qt) Dq {f (t)} − f (t) Dq {g(t)} f (t) = , g(t) g(t) g(q t)

(4)

so that, clearly,        g(q t) Dq f (t) − f (q t) Dq g(t) d f (t) f (t) = lim = . lim Dq q→1 q→1 g(t) g(t) g(q t) dt g(t)

(5)

It is known (see [618, pp. 3–4]) that there does not exist a general chain rule for q-derivatives. There is an exception given by
Dq {f (u(x))} = Dqβ f (u(x)) · Dq {u(x)},

(6)

where u = u(x) = α xβ , α and β being constants.

q-Antiderivative and Jackson q-Integral The function F(t) is a q-antiderivative of f (t), if Dq {F(t)} = f (t). It is denoted by Z

f (t) dq t.

(7)

In ordinary calculus, an antiderivative is unique up to an additive constant. However, in the case of a q-antiderivative, it is known (see [618, p. 65, Proposition 18.1]) that, for 0 < q < 1, up to an additive constant, any function f (t) has at most one q-antiderivative that is continuous at t = 0.

q-Extensions of Some Special Functions and Polynomials

485

The Jackson integral of f (t) is, thus, defined, formally, by Z

∞ X

f (t) dq t := (1 − q)t


qj f qj t ,

(8)

j=0

which can be easily generalized as follows: Z

f (t) dq g(t) =

∞ X

f qj t

 

g q j t − g qj+1 t .

(9)

j=0

The following theorem gives a sufficient condition under which the formal series in (8) actually converges to a q-antiderivative. Theorem 6.1 (618, p. 68, Theorem 19.1) Suppose 0 < q < 1. If |f (t) tα | is bounded on the interval (0, A] for some 0 5 α < 1, then the Jackson integral, defined by (7), converges to a function F(t) on (0, A], which is a q-antiderivative of f (t). Moreover, F(t) is continuous at t = 0 with F(0) = 0. Suppose that 0 < a < b. The definite q-integral is defined as follows: Zb

f (t) dq t := (1 − q)b

∞ X

q j f (q j b)

(10)

j=0

0

and Zb a

f (t) dq t =

Zb

f (t) dq t −

0

Za

f (t) dq t.

(11)

0

A more general version of (10) is given by Zb

f (t) dq g(t) =

∞ X

 

f q j b g q j b − g qj+1 b .

(12)

j=0

0

The improper q-integral of f (t) on [0, ∞) is defined by Z∞ 0

f (t) dq t : =

∞ Zq X

j

f (t) dq t

j=−∞ j+1 q

= (1 − q)

∞ X j=−∞

(13) f (q j ) q j

(0 < q < 1)

486

Zeta and q-Zeta Functions and Associated Series and Integrals

and Z∞

j+1

f (t) dq t =

q ∞ Z X j=−∞

0

=

q−1 q

f (t) dq t

qj ∞ X

(14) f (q j ) q j

(q > 1).

j=−∞

As noted earlier (see [618, p. 71, Proposition 19.1]), the improper q-integral, defined above, converges, if tα f (t) is bounded in a neighborhood of t = 0 with some α < 1 and for sufficiently large t with some α > 1. The formula for q-integration by parts, the fundamental theorem of q-calculus and q-Taylor formula are given as follows (see [618, Chapter 20]): Zb

f (t) dq g(t) = f (b) g(b) − f (a) g(a) −

a

Zb

g(q t) dq f (t)

(0 5 a < b 5 ∞).

a

(15) Theorem 6.2 (Fundamental Theorem of q-Calculus) If F(t) is an antiderivative of f (t) and if F(t) is continuous at t = 0, then Zb

f (t) dq t = F(b) − F(a)

(0 5 a < b 5 ∞).

(16)

a

The q-analogue of (t − a)n is defined by the polynomial (t − a)nq :=



1 (n = 0) (t − a) (t − q a) · · · (t − qn−1 a) (n ∈ N).

(17)

It is easy to see that Dq {(t − a)nq } = [n]q (t − a)n−1 q

(n ∈ N).

(18)

Theorem 6.3 (q-Taylor Formula with the Cauchy Remainder Term) Suppose that j Dq {f (t)} is continuous at t = 0 for any j, n ∈ N0 ( j 5 n + 1). Then a q-analogue of Taylor’s formula with the Cauchy remainder is given as follows:

f (b) =

n  X j=0

Dqj f



j

(b − a)q 1 (a) + [j]q ! [n]q !

Zb a

n Dn+1 q {f (t)}(b − q t)q dq t.

(19)

q-Extensions of Some Special Functions and Polynomials

487

6.3 q-Binomial Theorem We begin by recalling the well-known Ramanujan’s 1 91 -sum:     q b ∞ ; q (q; q) ; q (az; q) X ∞ ∞ az a (a; q)k k ∞ ∞  z = 1 91 (a; b; q, z) := q
b (b; q)k (z; q) ; q (b; q) ; q ∞ az ∞ a k=−∞ ∞

(1)

∞

(|q| < 1; |a| > |q|; |b| < 1; |b/a| < |z| < 1). A simple proof of (1) is given in [584] (see also [1080]). A special case of (1) when b = q yields the q-binomial theorem: 1 80 (a; −; q, z) :=

∞ X (a; q)k k (az; q)∞ z = (q; q)k (z; q)∞

(|q| < 1; |z| < 1),

(2)

k=0

which were proven by several mathematicians, such as Cauchy [225] and Heine [549]. Two special cases of (2) when a = 0 and when z is replaced by z a−1 and a → ∞ yield Euler’s formulas: ∞ X k=0

zk 1 = (q; q)k (z; q)∞

(|q| < 1; |z| < 1)

(3)

and k ∞ X (−1)k q(2) k z = (z; q)∞ (q; q)k

(|q| < 1; |z| < 1),

(4)

k=0

respectively. It is observed that lim q↓1


(qa z; q)∞ = lim 1 80 qa ; −; q, z = 1 F0 (a; −; z) = (1 − z)−a q↓1 (z; q)∞ (|z| < 1; a ∈ C),

(5)

which, by the principle of analytic continuation, holds true for z ∈ C cut along the positive real axis from 1 to ∞, with (1 − z)−a positive when z is real and less than 1. The special case of (2) when a = q−n (n ∈ N0 ) gives 1 80 (q

−n

; −; q, z) = (z q

−n

; q)n = (−z) q

n −n(n+1)/2



 q ;q z n

(n ∈ N0 ),

(6)

which, by the principle of analytic continuation, holds true for z ∈ C and is seen to be equivalent to 6.1(9).

488

Zeta and q-Zeta Functions and Associated Series and Integrals

A q-analogue of the classical exponential function ez is defined by ezq :=

∞ X zk , [k]q !

(7)

k=0

and another q-analogue of the classical exponential function ez is defined by Eqz :=

∞ X

qk(k−1)/2

k=0

zk = (1 + (1 − q)z)∞ q . [k]q !

(8)

It is easily seen, by applying (3) and (4), that ezq Eq−z = 1.

(9)

From 6.1(37), we also see that ez1/q = Eqz .

(10)

These two q-exponential functions under q-differentiation are given by and Dq {Eqz } = Eqqz .

Dq {ezq } = ezq

(11)

In general, we have z+w ezq ew q 6= eq .

However, the following additive property does hold true: z+w ezq ew q = eq

(wz = qzw).

(12)

Corresponding to the above-defined q-exponential functions ezq and Eqz , the q-trigonometric functions are defined as follows: sinq x := cosq x :=

−ix eix q − eq

2i ix eq + e−ix q 2

and and

Sinq x :=

Eqix − Eq−ix

; 2i Eqix + Eq−ix Cosq x := . 2

(13) (14)

It is easy to see from (10) that Sinq x = sin1/q x

and

Cosq x = cos1/q x.

(15)

We can use (9) to obtain sinq x · Sinq x + cosq x · Cosq x = 1,

(16)

q-Extensions of Some Special Functions and Polynomials

489

which is the q-analogue of the elementary identity: sin2 x + cos2 x = 1. q-analogues of other trigonometric identities can also be obtained (see [468, p. 23]). We apply the chain rule 6.2(6) and use (11) to find the following derivative formulas for the q-trigonometric functions: Dq {sinq x} = cosq x

and Dq {Sinq x} = Cosq qx

(17)

and Dq {cosq x} = − sinq x

and Dq {Cosq x} = −Sinq x.

(18)

Ramanujan’s 1 91 -sum (1) includes the Jacobi triple product identity [602] as a limiting case: ∞ X

2

(−1)k qk zk = lim 1 91 (−1/c; 0; q2 , −qzc) c→0

k=−∞

(19)

    q  = zq; q2 ; q2 q2 ; q2 , ∞ z ∞ ∞ which can be used to express the theta functions as follows (see, e.g., Whittaker and Watson [1225, Chapter 21], Gasper and Rahman [468, Section 1.6] and Bellman [113]): θ1 (x) = 2 θ2 (x) = 2

∞ X k=0 ∞ X

(−1)k q(k+1/2) sin(2k + 1) x,

(20)

q(k+1/2) cos(2k + 1) x,

(21)

2

2

k=0

θ3 (x) = 1 + 2

∞ X

2

qk cos 2kx

(22)

k=1

and θ4 (x) = 1 + 2

∞ X

2

(−1)k qk cos 2kx.

(23)

k=1

In terms of infinite products, upon replacing z in (19) by q e2ix , −q e2ix , −e2ix and e2ix , respectively, we have θ1 (x) = 2 q1/4 sin x

∞    Y 1 − q2k 1 − 2 q2k cos 2x + q4k , k=1

(24)

490

Zeta and q-Zeta Functions and Associated Series and Integrals

θ2 (x) = 2 q1/4 cos x

∞    Y 1 − q2k 1 + 2 q2k cos 2x + q4k ,

(25)

k=1

θ3 (x) =

∞    Y 1 − q2k 1 + 2 q2k−1 cos 2x + q4k−2

(26)

k=1

and θ4 (x) =

∞    Y 1 − q2k 1 − 2 q2k−1 cos 2x + q4k−2 .

(27)

k=1

We conclude this section by giving a widely-investigated generalization r 8s of the function 1 80 (a; −; q, z) in (2), which is defined by   a1 , . . . , ar ; q, z = r 8s (a1 , . . . , ar ; b1 , . . . , bs ; q, z) r 8s  b1 , . . . , bs ; :=

∞ X

(−1)(1−r+s)k q(1−r+s)(2) k

k=0

·

(a1 ; q)k · · · (ar ; q)k zk , (b1 ; q)k · · · (bs ; q)k (q; q)k

(28)

provided that the generalized basic (or q-) hypergeometric series in (28) converges.

6.4 q-Gamma Function and q-Beta Function q-Gamma Function The classic Gamma function 0(z) (see Section 1.1) was found by Euler, while he was trying to extend the factorial n! = 0(n + 1) (n ∈ N0 ) to real numbers. The q-factorial function [n]q ! (n ∈ N0 ), defined by 6.1(26), can be rewritten as follows: (1 − q)

−n

∞ Y (q; q)∞ (1 − qk+1 ) = (1 − q)−n := 0q (n + 1) (1 − qk+1+n ) (qn+1 ; q)∞

(0 < q < 1).

k=0

(1) Replacing n by x − 1 in (1), Jackson [592] defined the q-Gamma function 0q (x) by 0q (x) :=

(q; q)∞ (1 − q)1−x (qx ; q)∞

(0 < q < 1).

(2)

Note that it is not completely obvious that Jackson’s q-Gamma function is the most natural extension of the above-defined 0q (n + 1) (n ∈ N). Askey [73] has found analogues of many of the known facts about the Gamma function, and these strongly

q-Extensions of Some Special Functions and Polynomials

491

indicate that (2) is the natural extension of 0q (n + 1) (n ∈ N), which reduces to 0(x) as q → 1. Also note that, throughout this section, 0 < q < 1 is assumed. Setting x = 1 in (2) gives 0q (1) = 1.

(3)

The q-Gamma function in (2) satisfies the fundamental functional relation 0(x + 1) = x 0(x) (see 1.1(9)): 0q (x + 1) =

1 − qx 0q (x) = [x]q 0q (x). 1−q

(4)

Indeed, by observing that (q x+1 ; q)∞ =

(qx ; q)∞ , 1 − qx

(5)

we find that (q; q)∞
(1 − q)−x q x+1 ; q ∞ 1 − qx 1 − qx (q; q)∞ 0q (x). (1 − q)1−x = = x 1 − q (q ; q)∞ 1−q

0q (x + 1) =

The q-Gamma function in (2) also satisfies the q-analogue of the Bohr-Mollerup theorem (see Theorem 1.1): Theorem 6.4 (73, Theorem 3.1) Let f (x) be a function that satisfies each of the following properties: (a) f (1) = 1; (b) for some q ∈ (0, 1), f (x + 1) =

1 − qx f (x); 1−q

(6)

(c) log f (x) is convex for x > 0.

Then, f (x) = 0q (x) =

(q; q)∞ (1 − q)1−x (qx ; q)∞

(x > 0).

Proof. It is proven from (3) and (4) that 0q (x) satisfies (a) and (b). Observe that, for 0 < q < 1, ∞
d X d log 0q (x) = − log(1 − q) − log 1 − qn+x dx dx n=0 ∞ X

= − log(1 − q) + log q

n=0

qn+x 1 − qn+x

492

Zeta and q-Zeta Functions and Associated Series and Integrals

and ∞

X qn+x d2 2 log 0 (x) = (log q) q
. n+x 2 dx2 n=0 1 − q Thus, log 0q (x) is convex for x > 0.



Let 0 < x < 1. The convexity of log f (x) implies log f (x + n) 5 log f (n) + x [log f (n + 1) − log f (n)] or f (n + x) 5 f (n)



1 − qn 1−q

x

.

The functional equation (6) then gives   (1 − q)n 1 − qn x (1 − q)n f (x + n) 5 x f (n) f (x) = x 1−q (q ; q)n (q ; q)n (1 − q)n (q; q)n−1 (1 − qn )x = x (q ; q)n (1 − q)n−1 (1 − q)x so f (x) 5


(q; q)n−1 1−x n x (1 − q) 1 − q (qx ; q)n

(0 < q < 1, 0 < x < 1).

(7)

To find a lower bound for f (x), apply the convexity of log f (x) at n + x, n + 1 and n + x + 1 to obtain log f (n + 1) 5 log f (n + x) + (1 − x) [log f (n + x + 1) − log f (n + x)] or f (n + 1) 5 f (n + x)



1 − qn+x 1−q

1−x

.

As above, the functional equation (6) gives (1 − q)n (1 − q)n f (x + n) = x f (x) = x (q ; q)n (q ; q)n =

(1 − q)1−x (q; q)n ,
1−x 1 − qn+x (qx ; q)n



1−q 1 − qn+x

1−x

f (n + 1)

q-Extensions of Some Special Functions and Polynomials

493

so f (x) =

(q; q)n (1 − q)1−x
1−x (qx ; q)n 1 − qn+x

(0 < q < 1, 0 < x < 1).

(8)

Setting n → ∞ in (7) and (8) gives f (x) =

(q; q)∞ (1 − q)1−x = 0q (x). (qx ; q)∞

The functional equation (6) then gives f (x) = 0q (x) for x > 0. Remark 1 The q-gamma function has simple poles at x = 0, −1, −2, . . . and the residues Res 0q (x) =

x=−n

(1 − q)n+1
q−n ; q n log q−1

(n ∈ N0 ).

(9)

Indeed, lim (x + n) 0q (x) = lim

x→−n

x→−n

(x + n) (q; q)∞
· · · (1 − q)1−x (1 − qx ) 1 − q x+1

=

(1 − q)n+1 x+n

lim 1 − q−n · · · 1 − q−1 x→−n 1 − q x+n

=

(1 − q)n+1
. q−n ; q n log q−1

The q-gamma function has no zeros, so its reciprocal is an entire function with zeros at x = −n (n ∈ N0 ). ∞ Y 1 1 − qn+x x−1 = (1 − q) , 0q (x) 1 − qn+1

(10)

n=0

which also has zeros at x = −n + (2πik/ log q) (k ∈ Z; n ∈ N0 ). Applying 6.3(3) and 6.3(4) to the definition of the q-Gamma function (2) yields 0q (x) = (q; q)∞ (1 − q)1−x

∞ X k=0

qkx (q; q)k

(11)

and k ∞ 1 (1 − q)x−1 X (−1)k q(2) qkx = . 0q (x) (q; q)∞ (q; q)k

k=0

(12)

494

Zeta and q-Zeta Functions and Associated Series and Integrals

  Now, the generalized binomial coefficient βα , defined by 6.1(33), can, in terms of q q-Gamma function, be rewritten like 1.1(41) as follows:   0q (α + 1) α = . (13) β q 0q (β + 1) 0q (α − β + 1) The q-factorial [n]q !, defined by 6.1(26), is a monotone increasing function of q for q > 0, so it is natural to ask if something happens for the q-Gamma function. Since 0q (1) = 0q (2) = 1, something different may happen for 1 < x < 2 than for other x > 0. Theorem 6.5 (73, Theorem 4.1) The following inequalities hold: 0r (x) 5 0q (x) 5 0(x)

(0 < x 5 1 or x = 2, 0 < r < q < 1),

(14)

0(x) 5 0q (x) 5 0r (x)

(1 5 x 5 2, 0 < r < q < 1).

(15)

Also lim 0q (x) = 0(x).

q→1−

(16)

Proof. Only the proof of (16) is provided. By (14) and (15), 0q (x) is a monotone function of q, which is bounded, and so has a limit. By Theorem 6.4, the limit satisfies the assumptions of the Bohr-Mollerup theorem (see Theorem 1.1) and so is 0(x).  A q-analogue of Legendre duplication formula for the Gamma function 1.1(29) is given as follows:     1 1 0q (2x) 0q2 = 0q2 (x) 0q2 x + (1 + q)2x−1 . (17) 2 2 Indeed, we have  


1−x+1/2−x 0q2 (x) 0q2 x + 21 q2 ; q2 ∞ q; q2 ∞ 1 − q2   =


1/2 q2x ; q2 ∞ q2x+1 ; q2 ∞ 1 − q2 0q2 21 1−2x (q; q)∞  = 2x
1 − q2 = 0q (2x) (1 + q)1−2x . q ;q ∞ Also, a q-analogue of Gauss multiplication formula for the Gamma function 1.1(51) is similarly proven:       1 2 n−1 0q (nx) 0qn 0qn · · · 0qn n n n     (18) 1 n−1 = 0qn (x) 0qn x + · · · 0qn x + [n]nx−1 (n ∈ N). q n n

q-Extensions of Some Special Functions and Polynomials

495

q-Beta Function A special case of 6.2(10) when b = 1 yields Z1

f (t) dq t = (1 − q)

∞ X


qn f qn .

(19)

n=0

0

Now, try to get a q-analogue of the Beta function, defined by 1.1(59), and keep the integrand f (t) = tα−1 (1 − t)β−1 of 1.1(59) in mind. Since qn already appears in (19), it is natural to use tα−1 in itself. Even though there is no hope of evaluating a sum that contains a factor that (1 − qn )β−1 , in view of 6.3(2) and 6.3(5), there seems to be an alternative of a q-analogue of (1 − t)β−1 as follows:

∞ X q1−β ; q n n t q1−β ; q ∞ t = . (20) (q; q)n (t; q)∞ n=0

(20) is almost the right q-analogue of (1 − t)β−1 . The one thing wrong is that the β−1 function being integrated in 1.1(39) is not (1 − t)β−1 but (1 − t)+ , since the range of integration in 1.1(39) stops at t = 1. The first point in the sequence t = qn , which lies to the right of t = 1, is t = q−1 . So, by shifting the variable t by t qβ , Askey [73, Eq. (5.7)] chose a natural candidate for the q-Beta function: Bq (α, β) : =

Z1

tα−1

0

= (1 − q)

(t q; q)∞ dq t (t qβ ; q)∞ ∞ X

qnα

n=0

(<(α) > 0; β ∈ C \ Z− 0) (21)


qn+1 ; q ∞
. qn+y ; q ∞

We use 6.1(5) and 6.3(2) to prove a relationship between the q-Gamma function and the q-Beta function (see 1.1(42)), which shows a natural choice in (21). Theorem 6.6 We have Bq (α, β) =

0q (α) 0q (β) . 0q (α + β)

(22)

Proof.
∞ (q; q)∞ X qβ ; q n nα
Bq (α, β) = (1 − q) β q q ; q ∞ n=0 (q; q)n
(1 − q) (q; q)∞ qα+β ; q ∞
= qβ ; q ∞ (qα ; q)∞ =

(q;q)∞ (qα ;q)∞

(1 − q)1−α (q;q)∞ (qα+β ;q)∞

(q;q)∞

(qβ ;q)∞

(1 − q)1−β

(1 − q)1−α−β

=

0q (α) 0q (β) . 0q (α + β) 

496

Zeta and q-Zeta Functions and Associated Series and Integrals

Although it is not possible to change variables in a sum (and so in a q-integral), there are times when a change of variables in an ordinary integral will lead to another integral that can be approximated by a q-integral. For example, setting u = ct in 1.1(41) yields B(x, y) = cx

Z∞ 0

tx−1 dt (1 + ct)x+y

(<(x) > 0; <(y) > 0).

(23)

Askey [73] observed that Ramanujan’s sum 6.3(1) gives a q-integral extension of (23). Indeed, we use 6.1(5) to rewrite 6.3(1) as follows:  
q b ∞ X (bqn ; q)∞ n (ax; q)∞ ax ; q ∞ (q; q)∞ a ; q ∞   x =
, b (aqn ; q)∞ (x; q)∞ ax ;q (a; q)∞ aq ; q ∞ n=−∞ ∞

α+β and using the q-binomial theowhich, upon setting x = qα , a = −c,
b = −cnq α+β+n rem 6.3(2) to replace −c q ; q ∞ / (−c q ; q)∞ , gives

"∞ # X (qα+β ; q)k n k (−c q ) qαn (q; q) k n=−∞ ∞ X

=

k=0 (−cqα ; q)∞ (−c−1 q1−α ; q)∞ (q; q)∞ (qα+β ; q)∞ . (−c; q)∞ (−c−1 q; q)∞ (qα ; q)∞ (qβ ; q)∞

(24)

If we use 6.2(13) and 6.3(2), (24) can be rewritten as Z∞ X ∞ 0

k=0

(qα+β ; q)k (−c x)k xα−1 dq x (q; q)k

(−cqα ; q)∞ (−c−1 q1−α ; q)∞ 0q (α) 0q (β) (−c; q)∞ (−c−1 q; q)∞ 0q (α + β) Z∞ (−cqα+β x; q)∞ α−1 = x dq x. (−cx; q)∞ =

0

In view of 6.3(5), it is seen that (−cqα ; q)∞ −c−1 q1−α ; q
lim q→1 (−c; q)∞ −c−1 q; q ∞


∞

= (1 + c)

and −cqα+β x; q




lim

q→1

(−cx; q)∞

∞

= (1 + cx)−α−β .

−α

  1 α 1+ = c−α c

(25)

q-Extensions of Some Special Functions and Polynomials

497

6.5 A q-Extension of the Multiple Gamma Functions Motivated by Theorem 6.4 and Theorem 1.5, Nishizawa [865, 866] and Ueno and Nishizawa [1176] presented a q-analogue of the multiple Gamma functions (see Section 1.4) and its properties. Here, we introduce their works without proof. We assume 0 < q < 1 throughout this section. We begin by giving q-analogues of Gauss’s and Euler’s product forms of the Gamma function 1.1(4) and 1.1(7) (see [865, Equations (3.2) and (3.3)]): 0q (z + 1) = lim

N→∞

[1][2] · · · [N] [N + 1]z [z + 1][z + 2] · · · [z + N]

(1)

and (    ) ∞  Y [n + 1] z [z + n] −1 0q (z + 1) = , [n] [n]

(2)

n=1

where [z] = [z]q throughout this section. Nishizawa [865, Definition 4.1] defines the q-analogue of Vigne´ ras’s Gr -function (see Theorems 1.4 and 1.5) as follows: Definition 6.7 For z ∈ C with <(z) > 0, G0 (z + 1; q) := [z + 1], 

z

Gr (z + 1; q) := (1 − q)− r

∞   Y

(−1) 1 − qz+n

 n=1 

1 − qn

r(n+r−2 r−1 )

(1 − qn )gr (z,n)

  

(r ∈ N),

 

where g1 (z, n) := 0, gr (z, n) :=

r−1 X k=1

(−1)k−1



z r−k



 n+k−2 (r ∈ N \ {1}). k−1

We note that G1 (z; q) = 0q (z). The infinite products of these functions are absolutely convergent. Nishizawa [865, Theorem 4.2] (see also [866, Theorem 3.1]) proves a q-analogue of Theorem 1.5: Theorem 6.8 A unique hierarchy of functions exists, which satisies (i) Gr (z + 1; q) = Gr−1 (z; q) Gr (z; q), (ii) Gr (1; q) = 1, dr+1 (iii) dz r+1 log Gr+1 (z + 1; q) = 0 (z = 0), (iv) G0 (z; q) = [z],

498

Zeta and q-Zeta Functions and Associated Series and Integrals

where 

   z−u −u gr (z, u) = − . r−1 r−1 Gr (z + 1; q) is expressed as the following infinite product representation:   −k ∞  gr (z,k)  z+k (r−1)  Y 1 − q 1 − qk (r ∈ N). (3) Gr (z + 1; q) := (1 − q)−( )   1 − qk z r

k=1

The case of Theorem 6.8 when r = 1 corresponds to Askey’s theorem (see Theorem 6.4). So, the sequence {Gr (z; q)} includes a q-Gamma function. We call an element of the sequence a multiple q-Gamma function. The expression (3) can be regarded as a q-analogue of the Weierstrass product form for the function Gr (z) given in Theorem 1.4. Nishizawa [865, Proposition 4.4] (see also [866, Proposition 3.2]) derives the following counterpart of (1) and (2) for the function Gr (z; q): Theorem 6.9 If <(z) > 0, then Gr (z + 1; q) = lim

N→∞



Gr−1 (1; q) · · · Gr−1 (N; q) Gr−1 (z + 1; q) · · · Gr−1 (z + N; q) ) r Y z ( ) · Gr−m (N + 1; q) m m=1

and

Gr (z + 1; q) =

∞ Y n=1

(

 z ) r  Gr−1 (n; q) Y Gr−m (n + 1; q) (m) . Gr−1 (z + n; q) Gr−m (n; q) m=1

Koornwinder [694, Theorem B.2] proved rigorously the following result: lim 0q (z + 1) = 0(z + 1) q↑1

(z ∈ C \ {−1, −2, . . .}).

(4)

By making use of the Euler-Maclaurin summation formula (see 2.7(21)), Ueno and Nishizawa [1176, Proposition 4.1] (see also Nishizawa [866, Proposition 3.3]) obtain an expansion formula of log Gn (z + 1; q) as follows:

q-Extensions of Some Special Functions and Polynomials

499

Theorem 6.10 Suppose that <(z) > −1 and m > n. They have (    )  X n 1 − qz+1 d r−1 z z+1 Br log log Gn (z + 1; q) = − + n r! dz n−1 1−q r=1

+

( n X r=1

+

n−1 X j=0

d − dz

r−1 

) Zz+1

z n−1

1

ξ r qξ log q dξ r! 1 − qξ

(5)

m X Br Gn,j (z) Cj (q) + Fn,r−1 (z; q) − Rn,m (z; q), r! r=1

where   −t 1 − qz+t log , n−1 1 − qz+1 t=1 Z∞ n+1 X (−1)n e Bn+1 (t) Br fj+1,r−1 (1; q) + fj+1,n+1 (t; q) dt, Cj (q) := − r! (n + 1)!

Fn,r−1 (z; q) :=



dr−1 dtr−1



r=1

dr−1

1

 1 − qt j fj+1,r−1 (t; q) := r−1 t log , 1−q dt    Z∞ −t 1 − qz+t (−1)m−1 dm e log dt. Rn,m (z; q) := Bm (t) m n−1 m! dt 1 − qz+1 

1

The formula (5) is a generalization of Moak’s [838, Theorem 2]. As remarked by Daalhuis [361], the formula (5) is not an asymptotic expansion. Yet, Ueno and Nishizawa [1176] (see also Nishizawa [866]) observes that each term of (5) converges uniformly as q ↑ 1. Thus, they obtain the following theorem (see Nishizawa [866, Theorem 3.4]): Theorem 6.11 As q → 1 − 0, Gn (z + 1; q) converges to Gn (z + 1) uniformly on any compact set in the domain C \ {−1, −2, . . .}.

6.6 q-Bernoulli Numbers and q-Bernoulli Polynomials Carlitz ([215] and [217]) introduced q-extensions of the classic Bernoulli numbers and polynomials. Since then, many authors have studied this and related subjects (see, e.g., [227, 228, 255, 363, 493, 626, 649, 652, 653, 659, 660, 682, 971, 992, 1006, 1100, 1168, 1170] and [1174]). In the frequently cited [215], Carlitz introduced the relation (qη + 1)k = ηk

(k ∈ N \ {1}),

η0 = 1,

η1 = 0

(1)

500

Zeta and q-Zeta Functions and Associated Series and Integrals

as an inductive definition for a certain sequence {ηk }∞ k=0 of functions ηk = ηk;q , depending rationally on the parameter q. This definition is interpreted according to the umbral calculus convention that, after expanding the left side into monomials, one replaces each power ηj by the corresponding sequence element ηj . The same paper exhibits polynomials ηk (x) = ηk;q (x) in qx , determined by a difference equation analogous to that governing the Bernoulli polynomials (see 1.7(4)), namely ηk;q (x + 1) − ηk;q (x) = k qx [x]k−1 q ,

ηk (0) = ηk ,

(2)

where [x]q is defined by 6.1(24). In what follows, we require, for convenience, that 0 < q < 1. Because ηk and ηk (x) do not remain finite at q = 1, Carlitz [215] used ηk and ηk (x) in (1) and (2) to define a set of numbers βk = βk;q , by βk = ηk + (q − 1) ηk+1 ,

(3)

and a set of polynomials βk (x) = βk;q (x), by the recurrence relation qx βk (x) = ηk (x) + (q − 1) ηk+1 (x),

βk := βk (0)

(k ∈ N0 ).

(4)

In the limit q → 1, βk reduces to the Bernoulli number Bk . These numbers βk;q and polynomials βk;q (x) are q-analogues of the ordinary Bernoulli numbers Bk and polynomials Bk (x) (see Section 1.6). Recently, a variety of works on q-Bernoulli numbers and polynomials and q-Euler numbers and polynomials have appeared, for example, Cenkci and Can [227], Ryoo [992] and Kim [654]. Kim, in particular, has published 50 papers on this and related subjects; a few are cited here. Although many authors, including those whose works are mentioned above, have treated more general generating functions of q-eta and q-epsilon polynomials than those discussed here, we, first, revisit the seminal work of Carlitz [215], substituting formal generating series for the difference equations employed by him (see (1.2)). In fact, Carlitz himself noted [215, p. 988, lines 4–6] that it is easy to define numbers and polynomials, as well as generating functions, of higher order. We do propose certain minor corrections to [215]; corrected statements appear here, for example, as Equations (74) and (80). Here, we will recover the identities for ηk and ηk (x) given by Carlitz [215]. In place of the difference relations (1) and (2), defining ηk and ηk (x), we apply the (nowstandard) formal generating series Gq (x, t) := −t

∞ X

qk+x e[k+x]q t ≡

k=0

∞ X

ηk;q (x)

k=1

tk k!

(5)

and its specialization Gq (t) := Gq (0, t) = −t

∞ X k=0

qk e[k]q t ≡

∞ X k=1

ηk;q

tk . k!

(6)

q-Extensions of Some Special Functions and Polynomials

501

It is evident from (5) and (6) that ηk;q = ηk;q (0)

(k ∈ N).

(7)

To retain as close as possible a correspondence with Carlitz [215], we observe that the absence of any term involving η0;q in the generating series permits us to define η0;q = 1 as in (1). It follows from 6.1(24) that [x + y]q = [x]q + qx [y]q .

(8)

Using (8), we find that ηk;q (x) =

k   X k j=0

j

xj [x]k−j q q ηj;q

(k ∈ N0 ).

(9)

Indeed, by (5), (6) and (8), we have ∞ X

∞

ηk;q (x)

k=0

X tk x = −t qk+x e([x]q +[k]q q ) t k! k=0

[x]q t

=e

x

−q t

∞ X

! k [k]q qx t

q e

k=0

   ∞ ∞ j j X X t t ηj;q q xj  = e[x]q t Gq qx t =  [x]qj  j! j! j=0 j=0   k ∞ X X 1 xj   tk , [x]k−j = q ηj;q q j!(k − j)!


k=0

j=0

which, upon equating the coefficients of tk , yields (9). From (5), we get q-difference equation for Gx;q (t): et Gx;q (qt) = t qx e[x]q t + Gx;q (t),

(10)

which easily yields  k+1  X k+1 j=0

j

q j ηj;q (x) = (k + 1) qx [x]kq + ηk+1;q (x)

(k ∈ N0 ).

(11)

The special cases of (10) and (11) when x = 0, respectively, give the q-difference equation for Gq (t): et Gq (qt) = Gq (t) + t

(12)

502

Zeta and q-Zeta Functions and Associated Series and Integrals

and

ηk;q =

 1    q−1 k

if k = 1, (13)

k
j j ηj;q q

P   

j=0

if k ∈ N \ {1}.

We find from (9) and (13) that ηk;q (1) = ηk;q (0) = ηk;q

(k ∈ N \ {1}).

(14)

It is seen from (5) that Gx+1;q (t) − Gx;q (t) = t qx e[x]q t ,

(15)

which gives ηk+1;q (x + 1) − ηk+1;q (x) = (k + 1) qx [x]kq

(k ∈ N0 ).

(16)

Considering t

qj+x

e[j+x]q t = e 1−q e q−1

t

(17)

in (5), we find that ∞ X

ηk;q (x)

∞ ∞ ∞ X tk q(`+1)x t` X (`+1)j X (−1)` t` = −t q k! (q − 1)` `! (q − 1)` `! `=0

k=0

`=0

j=0

q(`+1)x t`

∞ X (−1)` t`
= −t 1 − q`+1 (q − 1)` `! `=0 (q − 1)` `! `=0   ∞ X k  ( j+1)x X q tk
(−1)k−j = −t ,  j!(k − j)! 1 − qj+1  (q − 1)k ∞ X

k=0

j=0

which, upon equating the coefficients of tk+1 and assuming 0/[0]q = 1, after a little simplification, yields an explicit expression of ηk;q (x): (q − 1)k ηk;q (x) =

k X

(−1)k−j

j=0

  k j jx q j [j]q

(k ∈ N),

(18)

which can be equivalently written as: k   X k j=0

j

(q − 1)j ηj;q (x) =

k kx q [k]q

(k ∈ N),

(19)

q-Extensions of Some Special Functions and Polynomials

503

by using the inverse relation technique in combinatorial analysis. Indeed, using a wellknown series manipulation k X ` X

k X k X

A`,j

(20)

      k ` k k−j = , ` j j `−j

(21)

A`,j =

j=0 `=j

`=0 j=0

and a binomial identity

we have k   X k `=0

  k k   X k j jx X `−j k − j q (−1) (q − 1) η`;q (x) = `−j ` j [j]q `

`=j

j=0

=

k  X j=0



k kx k j jx q (1 − 1)k−j = q . [k]q j [j]q

The special cases of (18) and (19) when x = 0, respectively, gives (q − 1) ηk;q = k

k X

(−1)

k−j

j=0

  k j j [j]q

(k ∈ N),

(22)

or, equivalently, k   X k j=0

j

(q − 1)j ηj;q =

k [k]q

(k ∈ N).

(23)

We give multiplication formula for ηk;q (x): [m]k−1 q

m−1 X

ηk;qm



η

  j = ηk;q (mx) x+ m

j=0

x+j m



= ηk;q (x)

(k ∈ N0 ; m ∈ N),

(24)

or, equivalently, [m]k−1 q

m−1 X j=0

k;qm

(k ∈ N0 ; m ∈ N).

(25)

504

Zeta and q-Zeta Functions and Associated Series and Integrals

Indeed, we have ∞ X

[m]k−1 q

m−1 X

ηk;qm



j=0

k=0

x+j m



  m−1 ∞ [m]q t tk 1 XX x+j = ηk;qm k! [m]q m k!


k

j=0 k=0

m−1 ∞ XX

= −t

qms+j+x e[ms+j+x]q t

j=0 s=0 ∞ X `+x [`+x]q t

= −t

q

e

`=0

=

∞ X

ηk;q (x)

k=0

tk . k!

It follows from (5) and (8) that
Gx+y;q (t) = e[y]q t Gx+y;q qy t ,

(26)

which yields ηk;q (x + y) =

k   X k j=0

j

If we replace q and x by

ηj;q (x) qyj [y]k−j q 1 q

(k ∈ N0 ).

(27)

and 1 − x, respectively, in (18) and use 6.1(35), we get

ηk;q−1 (1 − x) = (−1)k qk−1 ηk;q (x)

(k ∈ N),

(28)

the special case x = 1 of which, in view of (12), yields ηk;q−1 (0) = (−1)k qk−1 ηk;q (1) = (−1)k qk−1 ηk;q

(k ∈ N \ {1}).

(29)

q-Stirling Numbers of the Second Kind From the initial condition 10 f (x) = f (x) and the recurrence relation 1n+1 f (x) = 1n f (x + 1) − qn 1n f (x)

(n ∈ N0 ),

(30)

one may define (see [593], [869, Chapter 1] and [1209]) a sequence {1n }∞ n=0 of q-difference operators in which the index identifies the position of the pertinent operator within the sequence but is not directly interpreted as an operator exponent. It is easy to prove that 1n f (x) =

n X

1

(−1)j q 2 j( j−1)

j=0

where

n j q

is defined by 6.1(29).

  n f (x + n − j), j q

(31)

q-Extensions of Some Special Functions and Polynomials

505

If f (x) is a polynomial in qx of degree 5 n, it is obvious that we may put f (x) =

n X

αj [x]q;j ,

(32)

j=0

where [x]q;j is defined by 6.1(30) and αj is independent of x. To determine the coefficients αj , by making use of the easily proved formula 1n [x]q;j = [j]q;n [x]q;j−n qn(x−j+n) ,

(33)

apply 1n to both members of (32) and put x = 0 to finally get αj =

1j f (0) . [j]q !

(34)

q-Stirling numbers of the second kind Sq (n, j) are defined by taking f (x) = [x]nq in (32) and writing, for convenience, as [x]nq =

n X

1

q 2 j( j−1) Sq (n, j) [x]q;j ,

(35)

j=0

where Sq (n, j) is independent of x. It follows from (35) that
1 q 2 j( j−1) Sq (n, j) [x]q;j [x − j]q q j + [j]q X 1
= q 2 j( j−1) Sq (n, j − 1) + [j]q Sq (n, j) [x]q;j ,

[x]n+1 = q

X

which yields a recursion formula Sq (n + 1, j) = Sq (n, j − 1) + [j]q Sq (n, j).

(36)

It follows immediately from (36) that Sq (n, j) is a polynomial in q with integral coefficients. We use (32) and (31) to express Sq (n, j) explicitly: Sq (n, j) =

  1 j 1 q− 2 j( j−1) X j (−1)` q 2 `(`−1) [j − `]nq [j]q ! ` q

(n, j ∈ N0 ),

(37)

`=0

the right side of which vanishes for n < j and is equivalently written as   j 1 X j−` 21 `(`+1−2j) j Sq (n, j) = (−1) q [`]n [j]q ! ` q q `=0

(n, j ∈ N0 ).

(38)

506

Zeta and q-Zeta Functions and Associated Series and Integrals

Note that, upon taking the limit q → 1 on (38), Sq (n, j) reduces to the familiar Stirling numbers of the second kind (see 1.6(21)). A formula of a different sort may also be introduced:   n   X n n = (q − 1)`−j Sq (`, j), j q `

(39)

`=j

or, equivalently, (q − 1)n−j Sq (n, j) =

n X

(−1)n−`

`=j

   n ` . ` j q

(40)

Note that (38) and (40) are, in fact, equivalent. We define a slightly-generalized form of Sq (n, j) by means of [x + y]nq =

n X

q 2 (y+j)(y+j−1) Sq (n, j)(y) [x]q;j , 1

(41)

j=0

so that Sq (n, j)(0) = Sq (n, j). As in getting (36) and (37), we obtain a recursion formula Sq (n + 1, j)(y) = Sq (n, j − 1)(y) + [y + j]q Sq (n, j)(y),

(42)

which shows that Sq (n, j)(y) is a polynomial in qy ; an explicit formula q

1 2 (y+j)(y+j−1)

  j 1 X ` 21 `(`−1) j (−1) q Sq (n, j)(y) = [y + j − `]nq . [j]q ! ` q

(43)

`=0

It is easily seen that 1

q 2 y(y−1+2j) Sq (n, j)(y) =

n   X n

`

`=j

`y [y]n−` q q Sq (`, j).

(44)

It follows from either (43) or (44) that q

1 2 y(y−1)

n   X n

`

`=j

`−j

(q − 1)

Sq (`, j)(y) = q

(n−j)y

  n . j q

(45)

The Polynomial βk (x) = βk;q (x)  ∞ Let aj j=0 be an arbitrary sequence of numbers and define fn (x) =

n   X n j=0

j

jx aj [x]n−j q q .

(46)

q-Extensions of Some Special Functions and Polynomials

507

Then, fn (0) = an and fn (x + y) =

n   X n

j

j=0

jx fj (y) [x]n−j q q ,

(47)

the special case y = 0 of which reduces to (46). The polynomial ηn (x) = ηn;q (x) and the number ηn = ηn;q (0) do not remain finite when q = 1. Carlitz [215], therefore, introduces a polynomial βn (x) = βn;q (x) and number βn = βn;q that will approach a finite limit for q = 1. We, first, define βn = ηn + (q − 1) ηn+1

(n ∈ N0 ).

(48)

Repeated application of (48) leads to k   X k j=0

j

(q − 1) ηn−k+j = j

 k−1  X k−1 j

j=0

(q − 1)j βn−k+j ,

which, for n = k, becomes n   X n j=0

j

(q − 1) ηj = j

 n−1  X n−1 j

j=0

(q − 1)j βj ,

which, upon comparing with (23) and (22), leads at once to n   X n j=0

j

(q − 1) j βj =

n+1 [n + 1]q

(49)

and (q − 1)n βn =

n X

  n j+1 . j [j + 1]q

(−1)n−j

j=0

(50)

Carlitz [215] defines (q − 1) βn (x) = n

n X j=0

(−1)

n−j

  n j + 1 jx q , j [j + 1]q

(51)

which implies (4). It follows from (50) and (51) that βn (x) =

n   X n j=0

j


n x βj qjx [x]n−j q := q β + [x]q ,

(52)

508

Zeta and q-Zeta Functions and Associated Series and Integrals

which, in view of (47), generalizes the following result: βn (x + y) =

n   X n j=0

j


n x βj (y) qjx [x]n−j q := q β(y) + [x]q .

(53)

Analogous to (16), we find that q x+1 βn (x + 1) − q x βn (x) = n qx [x]n−1 + (q − 1) (n + 1) qx [x]nq , q so that q βn (x + 1) − βn (x) = n qx [x]n−1 + (q − 1) [x]nq , q which, for x = 0, becomes ( q(qβ + 1)n − βn =

1 0

(n = 1), (n ∈ N \ {1}).

(54)

(55)

Analogous to (25) and (28), we have [k]n−1 q

k−1 X j=0

  j = βn;q (kx) q j βn;qk x + k

(k ∈ N)

(56)

and βn;q−1 (1 − x) = (−q)n βn;q (x),

(57)

which, for x = 1, implies βn;q−1 (0) = (−q)n βn;q (1) = (−1)n qn−1 βn

(n ∈ N \ {1}).

(58)

It follows from (35) that k−1 X `=0

q` [`]nq =

n X

1

q 2 j( j+1) Sq (n, j)

j=0

[k]q;j+1 . [j + 1]q

(59)

Conversely, in view of (16), the left member of (59) is  n+1  1 1 X n+1 [k]qj q(n+1−j)k ηn+1−j (ηn+1 (k) − ηn+1 ) = n+1 n+1 j j=1  η n+1 (n+1)k + q −1 n+1 n    η X n n+1 (n−j)k ηn−j = [k]j+1 + q(n+1)k − 1 . q q j j+1 n+1 j=0

q-Extensions of Some Special Functions and Polynomials

509

Comparison with (59) yields an identity in k. Considering [k]q;j+1 = [k]q [k − 1]q;j

and

lim

k→0

q(n+1)k − 1 = (n + 1)(q − 1), [k]q

to divide both members of this identity by [k]q and then let k → 0, we obtain ηn + (q − 1)ηn+1 =

n X

1

q 2 j( j+1) Sq (n, j)

j=0

[−1]q;j , [j + 1]q

which, in view of (48), yields βn =

n X

(−1)j Sq (n, j)

j=0

[j]q ! . [j + 1]q

(60)

It is obvious from (60) and (37) that βn remains finite for q = 1. Indeed, lim βn =

q→1

n X j=0

  j j 1 X (−1)j−` ( j − `)n = Bn , j+1 `

(61)

`=0

where Bn is the n-th Bernoulli number in No¨ rlund’s notation (see 1.5(30)). Substitution from (37) in (60) leads to the explicit formula βn =

n X j=0

  j X 1 ` 12 `(`+1−2j) j (−1) q [`]n , [j + 1]q ` q q

(62)

`=0

which can, upon using (52), be more generally expressed as βn (x) =

n X j=0

  j X 1 ` 21 `(`+1−2j) j (−1) q [x + `]nq . [j + 1]q ` q

(63)

`=0

Comparison with (63) and (43) yields a generalized formula of (60): βn (x) =

n X

1

(−1)j q 2 x(x−1+2j) Sq (n, j)(x)

j=0

[j]q ! . [j + 1]q

(64)

6.7 q-Euler Numbers and q-Euler Polynomials We begin by defining a set polynomials k;q (x) = k (x) and a set of numbers k;q (0) = k;q = k by means of the following generating functions given by formal series: Hx;q (t) ≡ [2]q

∞ X j=0

(−1) q e

j j [j+x]q t

:=

∞ X k=0

k;q (x)

tk k!

(1)

510

Zeta and q-Zeta Functions and Associated Series and Integrals

and Hq (t) ≡ [2]q

∞ X

(−1)j q j e[j]q t :=

j=0

∞ X

k;q

k=0

tk . k!

(2)

It is easy to see from (1) that q Hx+1;q (t) + Hx;q (t) = [2]q e[x]q t ,

(3)

which immediately yields q k (x + 1) + k (x) = [2]q [x]kq

(k ∈ N0 ).

(4)

It is observed that k (x) is uniquely determined by the functional equation (4). From (4) follows a summation formula (−1)k−1 qk n (x + k) + n (x) = [2]q

k−1 X

(−1)j q j [x + j]nq .

(5)

j=0

Considering 6.6(17) in (1), analogous to 6.6(18), we obtain an explicit formula for k;q (x): (q − 1)k k;q (x) = (q + 1)

k X

(−1)k−j

j=0

  k q xj , j 1 + qj+1

(6)

or, equivalently, k   X k j=0

j

(q − 1)j j;q (x) =

(q + 1) qkx . 1 + qk+1

(7)

The special cases of (6) and (7) when x = 0, respectively, give (q − 1)k k;q =

k X

(−1)k−j

j=0

  k q+1 j 1 + qj+1

(8)

and k   X k j=0

j

(q − 1)j j;q =

q+1 . 1 + qk+1

(9)

Replacing x and q, respectively, by 1 − x and q−1 in (6) leads to k;q−1 (1 − x) = (−1)k qk k;q (x).

(10)

q-Extensions of Some Special Functions and Polynomials

511

From 6.6(8) we find that
Hx+y;q (t) = e[x]q t Hy;q qx t ,

(11)

which gives k;q (x + y) =

k   X k

j

j=0


k jx x [x]k−j q q j;q (y) := q (y) + [x]q .

(12)

The special case of (12) when y = 0 becomes k;q (x) =

k   X k j=0

j


k jx x [x]k−j q q j;q := q  + [x]q .

(13)

Rearranging the series in (1) as even and odd terms, we see that t qx Hx;q (t) = G x+1 ;q2 [(q + 1)t] − G x ;q2 [(q + 1)t],

(14)

2

2

which yields (k + 1) q k;q (x) = (q + 1) x

k+1

    x  x+1 ηk+1;q2 − ηk+1;q2 2 2

(k ∈ N0 ), (15)

serves to express qx k;q (x) in terms of η-polynomials. Corresponding to 6.6(25), we get the multiplication formulas [m]kq

m−1 X j=0

  qm + 1 j = (−q)j k;qm x + k;q (m x) m q+1

(m is odd)

(16)

and [2]q [m]kq

m−1 X

(−1)

j+1

η

k+1;qm

j=0

  j x+ = (k + 1)qmx k;q (m x) m

(m is even).

(17)

Note that, for m = 2, (17) reduces to (15). To get explicit formulas of a simpler kind, Carlitz [215] starts by noting the function (cf. [861, Chapter VII]) Yn;q (x) :=

n X k=0

  x qn−k
, j+1 +1 k q j=k q

(−1)n−k Qn

which is straightforwardly proven to satisfy   x q Yn;q (x + 1) + Yn;q (x) = . n q

(18)

(19)

512

Zeta and q-Zeta Functions and Associated Series and Integrals

In view of (4) and 6.6(35), we have n X

q n;q (x + 1) + n;q (x) = [2]q


1 q 2 k(k−1) Sq (n, k) [k]q ! q Yk;q (x + 1) + Yk;q (x) ,

k=0

which yields n;q (x) = [2]q

n X

1

q 2 k(k−1) Sq (n, k) [k]q ! Yk;q (x).

(20)

k=0

The special case of (20) when x = 0 implies n;q =

n X

1 Sq (n, k) [k]q ! (−1)k q 2 k(k+1) Qk
, j+1 + 1 j=1 q k=0

(21)

which, using 6.6(37), becomes n;q =

n X

k X

(−1)k qk Qk

k=0




qj+1 + 1

j=1

(−1)` q 2 `(`−1) 1

`=0

  k [k − `]nq . ` q

(22)

Conversely, if we use 6.6(41), we obtain q n;q (x + y + 1) + n;q (x + y) = [2]q

n X


1 q 2 (y+k)(y+k−1) Sq (n, k)(y) [k]q ! q Yk;q (x + 1) + Yk;q (x) ,

k=0

which gives n;q (x + y) = [2]q

n X

q 2 (y+k)(y+k−1) Sq (n, k)(y) [k]q ! Yk;q (x). 1

(23)

k=0

The special case of (23) when x = 0 becomes n;q (y) =

n X

1 Sq (n, k)(y) [k]q ! (−1)k q 2 (y+k)(y+k−1)+k Qk
, j+1 + 1 j=1 q k=0

(24)

which, upon using 6.6(43), gives n;q (y) =

n X

Qk k=0

k X

(−1)k qk j=1




qj+1 + 1

`=0

`

(−1) q

1 2 `(`−1)

  k [y + k − `]nq . ` q

(25)

q-Extensions of Some Special Functions and Polynomials

513

Remark 2 It is, in view of (22), observed that the product n   Y qj+1 + 1 n;q j=1

is a polynomial in q, which, for q = 1, reduces to the number Cn (see [869, p. 27]). Likewise, (25) indicates that the product −n

2

  n   Y 1 2( j+1) (q + 1) q + 1 n;q2 2 n

j=1

is a polynomial in q, which, for q = 1, reduces to the Euler number En .

6.8 The q-Apostol-Bernoulli Polynomials Bk(n) (x; λ) of Order n We begin this section by setting a = qn (n ∈ N) in the q-binomial theorem 6.3(2) to obtain 1 = n−1 (z; q)n Q

1

=

1 − z qk




∞ X [n]q;k k=0

[k]q !

zk .

(1)

k=0 (n)

A q-extension of the Apostol-Bernoulli polynomials Bk (x; λ; q) of order n ∈ N (see Section 1.7) is defined, here, by means of the following generating function: (n)

Gx;λ;q (t) ≡ (−t)n

∞ X [n]q;k k=0

:=

∞ X

[k]q !

λk qk+x e[k+x]q t

(n) Bk (x; λ; q)

k=0

(2)

tk , k! (n)

and a q-extension of the Apostol-Bernoulli numbers Bk (λ; q) of order n ∈ N is defined, here, by means of the following generating function: (n)

Gλ;q (t) ≡ (−t)n

∞ X [n]q;k k=0

:=

∞ X k=0

[k]q !

(n) Bk (λ; q)

λk qk e[k]q t

tk . k!

(3)

514

Zeta and q-Zeta Functions and Associated Series and Integrals

Remark 3 Here, for convenience, only the case of order α = n ∈ N is considered. It is (n) (n) easy to see that, when q → 1, the generating functions of Bk (x; λ; q) and Bk (λ; q) (n) (n) in (2) and (3) would tend, respectively, to those of Bk (x; λ) and Bk (λ) given by 1.8(13) and 1.8(14). It is also observed that (n)

(n)

Bk (0; λ; q) = Bk (λ; q)

(k ∈ N0 )

(4)

and (1)

Bk (x; λ; q) = Bk (x; λ; q)

(1)

and Bk (λ; q) = Bk (λ; q),

(5)

which were, very recently, introduced and investigated by Cenki and Can [227, Eqs. (4) and (6)]. We note that (1)

Bk (1; q) = Bk (q)

are the familiar q-Bernoulli numbers that were considered by many authors (see, e.g., [215], [228], [649], [659], [660], [682] and [1100]). (n) A q-extension of the Apostol-Euler polynomials Ek (x; λ; q) of order n ∈ N is defined, here, by means of the following generating function: (n)

Hx;λ;q (t) ≡ 2n

∞ X [n]q;k k=0

:=

∞ X

[k]q !

(−λ)k qk+x e[k+x]q t

(n) Ek (x; λ; q)

k=0

(6)

tk , k! (n)

and a q-extension of the Apostol-Euler numbers Ek (λ; q) of order n is defined here by means of the following generating function: (n)

Hλ;q (t) ≡ 2n

∞ X [n]q;k k=0

:=

∞ X

[k]q !

(−λ)k qk e[k]q t

(n) Ek (λ; q)

k=0

(7)

tk . k! (n)

Remark 4 It is easy to see that, when q → 1, the generating functions of Ek (x; λ; q) (n) (n) (n) and Ek (λ; q) in (6) and (7) would tend, respectively, to those of Ek (x; λ) and Ek (λ) given by 1.8(15) and 1.8(16). It is also observed that (n)

(n)

Ek (0; λ; q) = Ek (λ; q)

(k ∈ N0 ).

(8)

q-Extensions of Some Special Functions and Polynomials

515

We, first, examine (2) and (3) to get the following relationship between these two generating functions:
(n) (n) Gx;λ;q (t) = Gλ;q qx t e[x]q t q(1−n)x ,

(9)

which, upon considering (2) and (3) again, yields (n)

Bk (x; λ; q) =

k   X k

j

j=0

(n)

( j+1−n)x Bj (λ; q) [x]k−j q q

(k ∈ N0 ).

(10)

Next, by considering the following identity: t

k

q − 1−q t

e[k]q t = e 1−q e

,

we obtain (n)

t

Gλ;q (t) = (−t)n e 1−q t

= (−t)n e 1−q

∞ ∞ X [n]q;k λk qk X (−1)j qj k j t [k]q ! j! (1 − q)j j=0 ∞ (−1)j tj X [n]q;k [k]q ! j! (1 − q)j j=0 k=0

k=0 ∞ X

 k λ qj+1 .


In view of (1), the last sum is equal to 1/ λ qj+1 ; q n , and we, thus, find that ∞ X k=0

(n)

Bk (λ; q)

k ∞ X X 1 (−1)j tk tk
= (−t)n , k! (1 − q)k λ qj+1 ; q n j! (k − j)! j=0 k=0

which, upon equating the coefficients of tn+k on both sides, yields the following (n) explicit expression for Bk (λ; q): (n)

Bn+k (λ; q) = (−1)n

k   (k + 1)n X k (−1)j
k j λ qj+1 ; q n (1 − q) j=0

(k ∈ N0 ; n ∈ N).

(11)

In light of (5), a special case of (11) when n = 1 reduces to the following result: k   X 1 k (−1)j j Bk (λ; q) = j 1 − λ qj (1 − q)k−1

(k ∈ N0 ),

(12)

j=0

which provides a corrected version of the corresponding formula given in [227, p. 215].

516

Zeta and q-Zeta Functions and Associated Series and Integrals (n)

We, now, give the following q-difference equation for Gλ;q (t): (n)

(n)

(n−1)

λ et q1−n Gλ;q (qt) = Gλ;q (t) + t Gqλ;q (t)

(n ∈ N),

(13)

where (0)

Gqλ;q (t) := 1. Indeed, we have (n)

λ et q1−n Gλ;q (qt) = (−t)n

∞ X k=0

[n]q;k k k [k] t [k]q λ q e q, [n + k − 1]q [k]q !

which, upon considering the following relationship: [n − 1]q [k]q = 1− qk , [n + k − 1]q [n + k − 1]q leads us to (13). The special case of (13) when n = 1 reduces immediately to a known result [227, p. 216, Eq. (5)]. From (3) and (13) it is easy to derive the following q-difference equation for (n) Bk (λ; q): λ q1−n

 k+1  X k+1 j

j=0

(n)

(n−1)

(n)

q j Bj (λ; q) = Bk+1 (λ; q) + (k + 1) Bk

(qλ; q)

(14)

(k ∈ N0 ; n ∈ N). (n)

Remark 5 Since the q-difference equation for Gλ;q (t) in (13) holds true, if and only (n)

if the q-difference equation for Bk (λ; q) in (14) holds true and also since the coefficients of a power series are uniquely determined, we conclude that the generating (n) (n) function Gλ;q (t) of the q-extension of the Apostol-Bernoulli numbers Bk (λ; q) of (n)

order n can be determined as a solution of the q-difference equation for Gλ;q (t) in (13). (n)

Remark 6 The following q-distribution relation holds true for Bk (x; λ; q): [m]k−1 q

m−1 X j=0

λ

j

(n) Bk



 x+j m m (n) ; λ ; q = Bk (x; λ; q) m

(k ∈ N0 ; m, n ∈ N).

(15)

q-Extensions of Some Special Functions and Polynomials

517

The special case of (15) when n = 1 was proven by Cenki and Can [227, p. 216, Lemma 5]. Here we give an equivalence statement that (15) holds true for n ∈ N \ {1}. Indeed, we have (n;m) Ix;λ;q (t) ≡

∞ X

[m]k−1 q

=

1 [m]q 1 [m]q

λ

j

(n) Bk



j=0

k=0

=

m−1 X

m−1 X

λj

∞ X

j=0 m−1 X

(n)



Bk

k=0 (n)

λj G x+j m

j=0

;λm ;qm

x+j m m ;λ ;q m

x+j m m ;λ ;q m





tk k! [m]q t k!


k


[m]q t .

By applying (9) and the identity 6.5(8) to the last equation, we get (n;m)

Ix;λ;q (t) =

m−1
q(1−n)x e[x]q t X j (1−n)j (n) x λq Gλm ;qm q x+j [m]q t e[j]q q t . [m]q

(16)

j=0

Moreover, in view of (2) and (9), we have ∞ X

(n)

Bk (x; λ; q)

k=0


tk (n) = Gλ;q qx t e[x]q t q(1−n)x . k!

(17)

Now, it is easily seen from (16) and (17) that (15) holds true, if and only if the following result holds true: (n) Gλ;q (t) =

m−1
1 X j (1−n)j (n) λq Gλm ;qm qj [m]q t e[j]q t [m]q

(m, n ∈ N).

(18)

j=0

(n)

Let Pλ;q (t) be the right-hand side of (18). In view of Remark 5, it suffices to show that (n)

Pλ;q (t) satisfies the following q-difference equation: (n)

(n)

(n−1)

λ et q1−n Pλ;q (qt) = Pλ;q (t) + t Pqλ;q (t)

(n ∈ N \ {1}).

Indeed, if we begin with (n)

m
1 X j (1−n)j (n) λq Gλm ;qm qj [m]q t e[j]q t [m]q j=1 
1 (n) (n) = Pλ;q (t) + λm q(1−n)m Gλm ;qm qm [m]q t e[m]q t [m]q 
(n) − Gλm ;qm [m]q t

λ et q1−n Pλ;q (qt) =

(19)

518

Zeta and q-Zeta Functions and Associated Series and Integrals

and apply (13) to the last equation, we obtain
(n) (n) (n−1) λ et q1−n Pλ;q (qt) = Pλ;q (t) + t G(qλ)m ;qm [m]q t .

(20)

Now, it is seen from (19) and (20) that an equivalence condition for the validity of the (n) q-distribution relation for Bk (x; λ; q) in (15) can be written as in Remark 7 below. Remark 7 The q-difference equation (15) holds true, if and only if (n)

(n)

Pqλ;q (t) = G(qλ)m ;qm [m]q t




(m, n ∈ N),

that is, if and only if the following result holds true for n, m ∈ N: X
(n)
m−1
(n) [m]q − 1 G(qλ)m ;qm [m]q t = (qλ)j q(1−n)j G(qλ)m ;qm qj [m]q t e[j]q t ,

(21)

j=1

where an empty sum is understood (as usual) to be nil.

6.9 The q-Apostol-Euler Polynomials Ek(n) (x; λ) of Order n By applying the methodology and techniques used above in getting some identities for the generating functions of the q-extensions of the Apostol-Bernoulli polynomials and numbers, we can derive the following corresponding identities involving the generating functions of the q-extensions of the Apostol-Euler polynomials and numbers:
(n) (n) Hx;λ;q (t) = Hλ;q qx t e[x]q t qx . k   X k (n) (n) ( j+1)x Ek (x; λ; q) = E (λ; q) [x]k−j q q j j

(1) (k ∈ N0 ; n ∈ N).

(2)

j=0

(n) Ek (λ; q) =

k   X k (−1)j 1
j −λ qj+1 ; q n (1 − q)k j=0

(k ∈ N0 ; n ∈ N).

(3)

(n)

The q-difference equation for Hλ;q (t) is given by (n)

(n−1)

(n)

Hλ;q (t) − 2 Hqλ;q (t) = −λ et q Hλ;q (qt) where (0)

Hqλ;q (t) := 1.

(n ∈ N),

(4)

q-Extensions of Some Special Functions and Polynomials

519

(n)

The q-difference equation for Ek (λ; q) is given by (n−1) (n) Ek (λ; q) − 2 Ek (qλ; q) = −λ

k   X k j=0

j

(n)

Ej (λ; q) qj+1

(k ∈ N0 ; n ∈ N). (5)

(n)

(n)

The following relationship holds true between Bk (x; λ; q) and Ek (x; λ; q): (n)

(n)

(−2)n Bn+k (x; −λ; q) = (k + 1)n Ek (x; λ; q)

(k ∈ N0 ; n ∈ N),

(6)

which follows immediately, by using the following generating-function relationship: (n) Gx;−λ;q (t) =

  t n (n) − Hx;λ;q (t) 2

(n ∈ N).

(7)

Remark 8 We conjecture that the following q-distribution relation holds true for (n) Ek (x; λ; q): [m]k−1 q

m−1 X

(−λ)

j

j=0

(n) Ek



 x+j m m (n) ; λ ; q = Ek (x; λ; q) m

(k ∈ N0 ; m, n ∈ N). (8)

When n = 1, (8) is easily verified. The other cases of our conjectured relationship (8) when n ∈ N \ {1} remain to be proven.

6.10 A Generalized q-Zeta Function Many authors have tried to give q-analogues of the Riemann Zeta function ζ (s), defined by 2.3(1), and its related functions (see, e.g., [227, 255, 363, 493, 626, 652, 1006, 1100, 1168–1170] and [1174]). Here, we give a q-analogue of the generalized zeta function ζ (s, a), defined by 2.2(1), among other things, by just following the method of Kaneko et al. [626], who mainly used Euler-Maclaurin summation formula (see 2.7(21) or 2.7(22)) to present and investigate a q-analogue of the Riemann zeta function ζ (s), defined by 2.3(1), and gave a good and reasonable explanation that their q-analogue may be a best choice. They [626] also commented that q-analogue of ζ (s, a) can be achieved by modifying their method.

An Auxiliary Function Defining Generalized q-Zeta Function We begin this section by presenting a function gq (a; s, t), defined by gq (a; s, t) :=

∞ X q(n+a)t [n + a]sq n=0

(0 < q < 1; 0 < a 5 1).

(1)

520

Zeta and q-Zeta Functions and Associated Series and Integrals

Remark 1 It is easy to see that, by the ratio test, the series in (1) converges absolutely for <(t) > 0 and any s ∈ C. The series in (1) also converges to the generalized Zeta function ζ (s, a), defined by 2.2(1), as q ↑ 1, if <(s) > 1 and <(t) > 0. So, we can define gq (a; s, t) in (1) as a q-analogue of the generalized Zeta function ζ (s, a). However, Kaneko et al. [626] took several examples to contend that t = s − 1 in gq (1; s, t) seems the best possible choice to define a q-analogue of the Riemann Zeta function ζ (s), defined by 2.3(1). Theorem 6.12 For 0 < q < 1, as a function of (s, t) ∈ C2 , gq (a; s, t) in (1) is continued meromorphically, by means of the series expansion:  ∞  X s + r − 1 qa(t+r) s (2) gq (a; s, t) = (1 − q) 1 − qt+r r r=0

with simple poles at t = −r +

2`π log q

(r ∈ N0 ; ` ∈ Z).

(3)

Proof. It follows immediately from the binomial theorem that gq (a; s, t) = (1 − q)s

∞ X

q(n+a)t 1 − qn+a


−s

n=0

= (1 − q)s

∞ X

q(n+a)t

n=0

= (1 − q)s = (1 − q)s

 ∞  X s+r−1 r

r=0

 ∞ ∞  X s+r−1 X r=0 ∞  X r=0

r

q(n+a)r

q(n+a)(t+r)

n=0

s + r − 1 qa(t+r) . r 1 − qt+r 



In view of Remark 1, we also choose to define a q-analogue of the generalized Zeta function ζ (s, a) in 2.2(1) by ζq (s, a) := gq (a; s, s − 1) :=

∞ X q(n+a)(s−1) [n + a]sq

(0 < q < 1; 0 < a 5 1).

n=0

We list, here, some properties that easily follow from Theorem 6.12 as Theorem 6.13 (a) The function ζq (s, a) has simple poles at points in 1 + 2πi Z/ log q and in the set   2π i m `+ ` ∈ Z , m ∈ Z \ {0} , 50 log q

(4)

q-Extensions of Some Special Functions and Polynomials

521

where Z50 := Z \ N. In particular, s = 1 is a simple pole of ζq (s, a) with its residue (q − 1)/ log q. (b) For m ∈ N0 , the limiting value lim ζq (s, a) =: ζq (−m, a)

s→−m

exists and is given explicitly by ( ζq (−m, a) = (1 − q)

−m

m X r=0

)   (1−a)(m+1−r) (−1)m+1 m q (−1) + . (m + 1) log q r qm+1−r − 1

(5)

(m ∈ N0 ),

(6)

r

Now, we have Theorem 6.14 lim ζq (−m, a) = − q↑1

Bm+1 (a) m+1

where Bm+1 (a) denotes the Bernoulli polynomials (see Section 1.7). Proof. In view of (5), it is equivalent to show that

lim (1 − q)

−m

( m X

q↑1

r=0

)   (1−a)(m+1−r) m q (−1)m+1 Bm+1 (a) (−1) + =− , m+1−r r (m + 1) log q m+1 q −1 r

which, upon multiplying (−1)m+1 (m + 1) and setting m + 1 − r = r0 and then dropping the prime on r in the resulting equation, yields ( lim (1 − q) q↑1

−m

(m + 1)

m+1 X r=1

(−1)

r

)  (1−a)r 1 m q + = (−1)m Bm+1 (a). r − 1 qr − 1 log q



(7) Using 1.6(1) q(1−a)r e(1−a)r log q 1 r log q e(1−a)r log q 1 = r log q = r q −1 r log q e −1 er log q − 1 ∞ k 1X (r log q) 1 = Bk (1 − a) , r k! log q k=0

522

Zeta and q-Zeta Functions and Associated Series and Integrals

we have (m + 1)

m+1 X

(−1)

r

r=1

 (1−a)r m q r − 1 qr − 1



m+1 X

 X ∞ m 1 (r log q)k 1 Bk (1 − a) = (m + 1) (−1) r−1 r k! log q r=1 k=0 !   ∞ m+1 X X m+1 k (log q)k−1 = r Bk (1 − a) . (−1)r k! r r



r=1

k=0

The inner sum of the last expression can be evaluated as m+1 X

(−1)r



r=1

   o m+1 k d kn r = x (1 − x)m+1 − 1 x=1 r dx  (k = 0), −1 0 (0 < k < m + 1), =  m+1 (−1) (m + 1)! (k = m + 1),

and we find (m + 1)

m+1 X

(−1)

r

r=1

=−

 (1−a)r m q r − 1 qr − 1



  1 + (−1)m+1 Bm+1 (1 − a) (log q)m + O (log q)m+1 log q

(q → 1).

From this observation, the expansion   log q = q − 1 + O (q − 1)2 (q → 1) and the known relation 1.6(10), we prove the desired result (7): ( lim (1 − q)−m (m + 1)

q→1

m+1 X

(−1)r

r=1

= (−1)

m+1

Bm+1 (1 − a) lim

q→1



)  (1−a)r m q 1 + r − 1 qr − 1 log q

(log q)m = −Bm+1 (1 − a) = (−1)m Bm+1 (a). (1 − q)m 

Definition 6.8 In view of Theorem 6.14, it is natural to define the q-Bernoulli polynomials Bn;q (a) by Bn;q (a) := −n ζq (1 − n, a)

(n ∈ N).

(8)

q-Extensions of Some Special Functions and Polynomials

523

If we use (8), then we get some interesting properties and relations, which are summarized as in the following theorem: Theorem 6.15 (a) Bn;q (a) can be expressed in an explicit form: (q − 1)n Bn;q (a) =

n X

(−1)r

r=0

  n r (1−a)r q r [r]q

(n ∈ N),

(9)

where the term with r = 0 is understood to be lim

r→0

1 r = . qr − 1 log q

(10)

So B0;q (a) may be defined as B0;q (a) :=

q−1 . log q

(11)

(9) can be written equivalently as follows: n X r=0

(−1)r

  n n q(1−a)n . (q − 1)r Br;q (a) = [n]q r

(12)

(b) We have a relationship between the Carlitz’s q-Bernoulli polynomials βn (a) in 6.6(51) and Bn;q (a) here:  qa βn (a) = (−1)n Bn;q (1 − a) + (1 − q) Bn+1;q (1 − a)

(n ∈ N).

(13)

(c) We have a relationship between βk (a) and ζq (−k, a): For n ∈ N,  qa βn (a) = (−1)n (q − 1) (n + 1) ζq (−n, 1 − a) − n ζq (1 − n, 1 − a) .

(14)

(d) We have a relationship between n;q (2a) in Section 6.7 and ζq2 (−n, a):     1 q2a n;q (2a) = (−1)n (q + 1)n+1 ζq2 −n, − a − ζq2 (−n, 1 − a) 2

(n ∈ N0 ). (15)

Proof. In view of (5), Bn;q (a) can be expressed as ( n X

)   n r 1 (1−a)r Bn;q (a) = (q − 1) (−1) q + r qr − 1 log q r=1 ( n )   X n r q(1−a)r = (q − 1)−n+1 (−1)r (n ∈ N), r qr − 1 −n+1

r

r=0

which is seen equal to (9). As in getting 6.6(19), (9) can be written as (12).



524

Zeta and q-Zeta Functions and Associated Series and Integrals

Using the familiar binomial identity 

     n+1 n n = + , r r r−1

we find from 6.6(51) that (q − 1) βn (a) = (−1) q n

n −a

( n X

(−1)r

r=1 n+1 X

−

r=1

  n r ra q r [r]q

)   n + 1 r ra q , (−1) [r]q r r

which, in view of (9), yields the relationship (13). (14) follows easily from (8) and (13). Note that, if we also assume the convention of (10) in the term with j = 0 of Equation 6.6(18), it is easy to see from 6.6(18) and (9) the following relation: ηn;q (a) = (−1)n Bn;q (1 − a)

(n ∈ N0 ).

(16)

Now, an application of (8) and (16) to 6.7(15) yields (15). Remark 2 Substitution of (16) for (13) is seen to come back to the defining relation 6.6(4). Taking the limit on each side of (14) as q ↑ 1, together with (6), reduces immediately to a familiar identity 1.7(10). Replacing q and a by q2 and 14 , respectively, in (15) and letting q ↑ 1 in the resulting equation, together with (6), in view of Remark 1, we obtain        1 22n+1 3 1 En = 2 En = Bn+1 − Bn+1 2 n+1 4 4 n

(n ∈ N0 ),

(17)

which is a special case of the well-known relationship between Euler polynomials En (x) and Bernoulli polynomials Bn (x) (see [1094, p. 65, Eq. (60)]; see also [869, p. 25, Eq. (25)]).

Application of Euler-Maclaurin Summation Formula Here, we mainly show Theorem 6.16 lim ζq (s, a) = ζ (s, a). q↑1

(18)

q-Extensions of Some Special Functions and Polynomials

525

We present, for convenience, a special case of the Euler-Maclaurin summation formula 2.7(21) when a = 0, b = N ∈ N, K = M + 1 ∈ N: N X

f (n) =

n=0

ZN

f (x) dx +

M  X Bk+1  (k) 1 f (N) − f (k) (0) ( f (0) + f (N)) + 2 (k + 1)! k=1

0

(−1)M+1 − (M + 1)!

ZN

(19) e BM+1 (x) f (M+1) (x) dx,

0

where f ∈ C∞ [0, ∞), N ∈ N and M ∈ N0 . The Hurwitz (or generalized) Zeta function ζ (s, a), defined by 2.2(1), is restricted as ζ (s, a) :=

∞ X

(<(s) > 1; 0 < a 5 1).

(k + a)−s

(20)

k=0

Applying f (x) = (x + a)−s (<(s) > 1) to (19) and taking the limit of each side of the resulting equation as N → ∞ gives M

ζ (s, a) =

X Bk+1 a−s+1 1 + s+ (s)k a−s−k s−1 2a (k + 1)! k=1

(s)M+1 − (M + 1)!

Z∞

(21)

e BM+1 (x) (x + a)−s−M−1 dx.

0

Remark 3 It is observed that the integral involved in (21) converges for <(s) > −M. So, ζ (s, a) can be continued analytically to <(s) > −M. Since M can be arbitrarily large, ζ (s, a) can be continued analytically to the whole s-plane, except for an obvious simple pole at s = 1. The special case of (19) when M = 1 is given as N X n=0

f (n) =

ZN

f (x) dx +


1 1 f 0 (N) − f 0 (0) ( f (0) + f (N)) + 2 12

0

−

1 2

ZN

(22) e B2 (x) f (2) (x) dx.

0

Consider a function fq (x; s, a) :=

q(x+a)(s−1)
s 1 − q x+a

(<(s) > 1; 0 < a 5 1)

(23)

526

Zeta and q-Zeta Functions and Associated Series and Integrals

ready for application to (22) to compute the following: ∂ s − 1 + q x+a fq (x; s, a) = (log q) q(x+a)(s−1)
s+1 ; ∂x 1 − q x+a

x+a + 1 − q x+a 2 ∂2 2 (x+a)(s−1) s(s + 1) − 3s 1 − q ; fq (x; s, a) = (log q) q
s+2 ∂x2 1 − q x+a and, in general,
−s−2 ∂k pq (x; s, a), fq (x; s, a) = (log q)k q(x+a)(s−1) 1 − q x+a k ∂x where pq (x; s, a) is a polynomial in s and q x+a . Applying (23) to (22), taking the limit of the resulting equation as N → ∞ and considering Z∞ 0

Z∞

q(x+a)(s−1)
s dx = 1 − q x+a

q−x−a


s dx q−x−a − 1 0 "
1−s #∞ q−x−a − 1 qa(s−1) (1 − qa )1−s = =− , (s − 1) log q (s − 1) log q

(24)

0

we readily find that, for <(s) > 1, ∞ (n+a)(s − 1) X q qa(s − 1) (1 − q)s (1 − qa )1 − s
s = − ζq (s, a) = (1 − q) (s − 1) log q 1 − qn+a n=0 s

a (1 − q)s qa(s − 1) 1 a(s − 1) s s−1 + q − (log q)q (1 − q) − (1 − q)s 2 (1 − qa )s 12 (1 − qa )s+1

2 Z∞ s(s + 1) − 3s 1 − q x+a + 1 − q x+a (log q)2 e · B2 (x)q(x+a)(s − 1) dx.
s+2 2 1 − q x+a

+

0

(25) Remark 4 Unlike in the classic case represented by (21), the integral in (25) cannot be made to converge by simply choosing M sufficiently large, instead of M = 1, because (M+1) the presence of the factor q(x+a)(s−1) in fq (x; s, a) implies that necessary <(s) > 1. Therefore, in this case, we use the known Fourier expansion of the periodic Bernoulli polynomials (see [1225, Chapter IX, Miscellaneous Exercise 12]): e Bk (x) = −k!

X n∈Z\{0}

e2π inx . (2πin)k

(26)

q-Extensions of Some Special Functions and Polynomials

527

The equality in (26) is valid for all real numbers x when k = 2, in which case the sum is absolutely and uniformly convergent. Substituting (26) with k = 2 into (25) and interchanging the summation and integration, we find that qa(s−1) (1 − q)s (1 − qa )1−s (1 − q)s qa(s−1) + (s − 1) log q 2 (1 − qa )s s − 1 + qa 1 − (log q) qa(s−1) (1 − q)s 12 (1 − qa )s+1 X 1 + (1 − q)s (log q)2 In (x; s, a, q), (2πin)2

ζq (s, a) = −

(27)

n∈Z\{0}

where In (x; s, a, q) :=

Z∞

2π inx (x+a)(s−1)

e

q

0



2 s(s + 1) − 3s 1 − q x+a + 1 − q x+a dx.
s+2 1 − q x+a

By letting q x+a = u in In (x; s, a, q), we obtain e−2π ina  s(s + 1) Bqa (s + δn − 1, −s − 1) log q  − 3s Bqa (s + δn − 1, −s) + Bqa (s + δn − 1, −s + 1) ,

In (x; s, a, q) = −

(28)

where 2πi δ := log q

and Bt (α, β) =

Zt

uα−1 (1 − u)β−1 du (<(α) > 0)

(29)

0

is the incomplete Beta function given in 1.1(61). It follows from (27) and (28) that qa(s−1) (1 − q)s (1 − qa )1−s (1 − q)s qa(s−1) + (s − 1) log q 2 (1 − qa )s 1 s − 1 + qa − (log q)qa(s−1) (1 − q)s 12 (1 − qa )s+1 (30) X e−2π ina  s − (1 − q) log q s(s + 1)Bqa (s + δn − 1, −s − 1) (2πin)2 n∈Z\{0}  −3sBqa (s + δn − 1, −s) + Bqa (s + δn − 1, −s + 1) .

ζq (s, a) = −

528

Zeta and q-Zeta Functions and Associated Series and Integrals

Note that each of the incomplete Beta functions in (30) converges absolutely for <(s) > 1 and is uniformly bounded with respect to n. Indeed, we see that a

Bqa (s − 1 + δn, −s + ν) 5

Zq

|u|<(s−2) uδn |1 − u|<(−s+ν−1) du

0

(31)

a

Zq =

uσ −2 (1 − u)−σ +ν−1 du (n ∈ Z \ {0}; σ = <(s); ν = −1, 0, 1).

0

Hence, the sum in (30) converges absolutely. Repeated use of integration by parts yields the formula Bt (α, β) =

M−1 X

(−1)k−1

k=1

+ (−1)M−1

(1 − β)k−1 α+k−1 t (1 − t)β−k (α)k

(1 − β)M−1 Bt (α + M − 1, β − M + 1) (α)M

(M ∈ N \ {1}). (32)

Applying (32) to Bqa (s − 1 + δn, −s − 1), we have Bqa (s−1 + δn, −s−1) =

M−1 X

(−1)k−1

k=1


−s−1−k (2 + s)k−1 a(s+δn+k−2) q 1−qa (s + δn−1)k

(2 + s)M−1 (s + δn−1)M−1 · Bqa (s−2 + M + δn, −s−M), + (−1)M−1

which, in view of (31), allows us to carry out the analytic continuation of Bqa (s − 1 + δn, −s − 1) as a function of s into the region <(s) > 2 − M. From this, we have X n∈Z\{0}

=

e−2π ina s(s + 1) Bqa (s + δn − 1, −s − 1) (2πin)2

M−1 X k=1

(−1)k−1

X n∈Z\{0}

− (−1)M−1 log q


−s−1−k e−2π ina (s)k+1 qa(s+δn+k−2) 1 − qa 2 (2πin) (s − 1 + δn)k

X n∈Z\{0}

Z∞ · 0

(s)M+1 (2πin)2 (s + δn − 1)M−1

e2π inx q(x+a)(s−2+M) 1 − q x+a


−s−M−1

dx

q-Extensions of Some Special Functions and Polynomials

529

from which we obtain lim (1 − q)s log q q↑1

e−2π ina s(s + 1) Bqa (s + δn − 1, −s − 1) (2πin)2

X n∈Z\{0}

=

M−1 X

X

k=1 n∈Z\{0}

X

−

n∈Z\{0}

=−

M−1 X k=1

1 (s)k+1 (2πin)2 ak+s+1 (2πin)k

(s)M+1 (2πin)2

Z∞ 0

(33)

e2π inx dx M−1 (2πin) (x + a)M+s+1

Bk+2 (s)k+1 a−k−s−1 + (s)M+1 (k + 2)!

Z∞ e BM+1 (x) (x + a)−M−s−1 dx. (M + 1)! 0

We also have lim q↑1

(1 − q)s (1 − qa )1−s = −a1−s log q

and

lim q↑1

(log q) (1 − q)s (1 − qa )s+1

=−

1 . as+1

(34)

Taking the limit of each side of (30) as q ↑ 1 and considering (33) and (34), we obtain M

lim ζq (s, a) = q↑1

X Bk+1 1 a−s+1 + s+ (s)k a−s−k s−1 2a (k + 1)! k=1

(s)M+1 − (M + 1)!

Z∞

(35)

e BM+1 (x) (x + a)−s−M−1 dx,

0

where we have used the following results: lim (1 − q)s log q q↑1

X n∈Z\{0}

e−2π ina s Bqa (s + δn − 1, −s) = 0 (2πin)2

and lim (1 − q)s log q q↑1

X n∈Z\{0}

e−2π ina s Bqa (s + δn − 1, −s + 1) = 0. (2πin)2

Hence, it follows from (21) and (35) that (18) is proven. We conclude this section by noting that all identities and properties in this section, obtained by setting a = 1, are seen to recover those in Kaneko et al. [626].

530

Zeta and q-Zeta Functions and Associated Series and Integrals

6.11 Multiple q-Zeta Functions Zhao [1261] defined and investigated a q analogue of the multiple Zeta functions defined by 2.7(1) based on the work of Kaneko et al. [626], which was studied in Section 6.8. Here, we introduce Zhao’s work [1261]. It is known (see Section 2.7) that the multiple Zeta function ζ (s1 , . . . , sd ) can be extended to a meromorphic function on Cd \ Dd , where 

Dd := (s1 , . . . , sd

) ∈ Cd

 sd = 1, sd−1 + sd = 2, 1, −2m (m ∈ N0 ), , sj + · · · + sd ∈ Z5d−j+1 ( j 5 d − 2)

(1)

ZP denoting the set of integers satisfying any property P . To find a q-analogue of the multiple Zeta functions ζ (s1 , . . . , sd ), defined by 2.7(1), Zhao [1261] defines an auxiliary function of 2d complex variables s1 , . . . , sd , t1 , . . . , td ∈ C X

gq (s1 , . . . , sd ; t1 , . . . , td ) :=

0
qk1 t1 +···+kd td , [k1 ]s1 · · · [kd ]sd

(2)


which converges, if < tj + · · · + td > 0, for all j = 1, . . . , d. Zhao [1261] also defines a multiple q-Zeta function by specialization of gq : ζq (s1 , . . . , sd ) := gq (s1 , . . . , sd ; s1 − 1, . . . , sd − 1) X qk1 (s1 −1)+···+kd (sd −1) = . [k1 ]s1 · · · [kd ]sd

(3)

0
Note that the special case of (3) when d = 1 is the same as the q-analogue of the Riemann Zeta function ζ (s) given by Kaneko et al. [626] (see also Section 6.8). We give a notation Dd (q) by ( ) s = 1 + 2π i Z, s ∈ Z + 2π i Z , d = 6 0 d 50 log q log q Dd (q) := (s1 , . . . , sd ) ∈ Cd , (4) 2π i sj + · · · + sd ∈ Z5d−j+1 + log q Z ( j < d) the last condition of which is vacuous, if d = 1.

Analytic Continuation of gq and ζq We start with a simple lemma: Lemma 6.17 For all 15

1 2

< q < 1, we have

1 − qk = [k]q < 2k 1−q

(k ∈ N).

Proof. Since 0 < qk 5 q < 1, the first inequality is obvious.



q-Extensions of Some Special Functions and Polynomials

531


Let f (q) = 2k(1 − q) − 1 − qk . We find that f (q) = 0 for all 0 < q < 1. Indeed, for all such q, f 0 (q) = −2k + k qk−1 5 −2k + k = −k < 0. So, f is strictly decreasing, and the positivity of f follows from f (1) = 0. This implies that [k]q < 2k as desired. Theorem 6.18 The function gq (s1 , . . . , sd ; t1 , . . . , td ) in (2) converges, if <(tj + · · · + td ) > 0, for all j = 1, . . . , d. It can be analytically continued to a meromorphic function over C2d , by means of the series expansion gq (s1 , . . . , sd ; t1 , . . . , td ) ∞ X

wt(Es)

= (1 − q)

# "  d Y sj + rj − 1 qj(rj +tj ) , 1 − qrj +tj +···+rd +td rj

(5)

r1 , ..., rd =0 j=1

where wt(Es) := s1 + · · · + sd . It has simple poles at tj + · · · + td ∈ Z50 +

2π i Z. log q

Proof. Assume <(sj ) < Nj ∈ N and let τj := <(tj ) for all j = 1, . . . , d. By Lemma 6.16, we have X

gq (s1 , . . . , sd ; t1 , . . . , td ) <

d Y

2 kj


Nj

qkj τj .

(6)

0
For simplicity, let P k = kd−1 (when d = 1, take k0 = 0), n = kd , N = Nd and τ = τd . Then, by root test, n>k nN qn τ converges, and moreover, X n>k

N nτ

n q

1 = N τ



d q dq

N X

qn τ = 1 − qτ


−N

fN (k; q, τ ) qk τ ,

(7)

n>k

P ` where fN (x; q, τ ) = N `=0 c` x is a polynomial of degree N, whose coefficients depend only on the constants N, q and τ . This proves the first part of the theorem when d = 1.  In the general case, it follows from (6) and (7) that N N1 +···+Nd X gq (s1 , . . . , sd ; t1 , . . . , td ) < 2 c` (1 − qτ )N `=0   d−2 X Y
N  · kj j qkj τj  kNd−1+` qk(τd−1 +τd ) . 0
j=1

532

Zeta and q-Zeta Functions and Associated Series and Integrals

Hence, the first part of the theorem follows from an easy induction on d. By using the binomial expansion for (1 − x)−s , we get gq (s1 , . . . , sd ; t1 , . . . , td ) = (1 − q)wt(Es)

∞ X

X

0
   d  Y s + r − 1 j j  qkj (rj +tj ) . rj j=1

As 0 < q < 1, the series converges absolutely by Stirling’s formula. So, we can exchange the summations. The theorem follows immediately from the next lemma, by taking zj = qrj +tj for j = 1, . . . , d. Lemma 6.19 Let zj ∈ C, such that |zj | < 1 for j = 1, . . . , d. Then, we find that X

d Y

k

zj j =

0
d Y j=1

zj · · · zd . 1 − zj · · · zd

(8)

Proof. When d = 1, this is clear. For d = 2, we denote the right hand side of (8), by Fd (z1 , . . . , zd ). Then, we have X

d Y

k

0
=

X

zj j =

X

d−1 Y

0
0
kd−1 +1 k j zd zj 1 − zd

=

k

X

zj j

k

zdd

kd−1
zd Fd−1 (z1 , . . . , zd−2 , zd−1 zd ). 1 − zd

The lemma follows immediately by induction.



Recalling ζq (s1 , . . . , sd ) = gq (s1 , . . . , sd ; s1 − 1, . . . , sd − 1), we have the following immediate consequence: Theorem 6.20 The q-multiple Zeta function ζq (s1 , . . . , sd ), defined by (3), can be extended to a meromorphic function with simple poles lying along Dd (q): ζq (s1 , . . . , sd ) wt(Es)

= (1 − q)

∞ X

# "  d Y sj + rj − 1 qj(rj +sj −1) . rj 1 − qrj +sj +···+rd +sd −d+j−1

(9)

r1 , ..., rd =0 j=1

To see the effect of taking different specializations of tj in gq , we define the shifting operators Sj (1 5 j 5 d) on the multiple Zeta function as follows:
Sj ζq (s1 , . . . , sd ) = ζq (s1 , . . . , sd ) + (1 − q) ζq s1 , . . . , sj−1 , . . . , sd . (10) It is obvious that these operators are commutative.

q-Extensions of Some Special Functions and Polynomials

533

Theorem 6.21 Let n1 , . . . , nd be nonnegative integers. Then, we have n

gq (s1 , . . . , sd ; s1 − 1 − n1 , . . . , sd − 1 − nd ) = Sn11 ◦ · · · ◦ Sdd ζq (s1 , . . . , sd )   nd n1 d   X X Y nj  = ··· (1 − q)rj  ζq (s1 − r1 , . . . , sd − rd ). rj r1 =0

rd =0

j=1

Proof. We sketch the proof only in the case n1 = · · · = nd−1 = 0. The general case is completely similar. In the rest of this section, we always let S be the shifting operator on the last variable. Suppose nd = n = 1. Then, we find gq (s1 , . . . , sd ; s1 − 1, . . . , sd−1 − 1, sd − 2) X qk1 (s1 −1)+···+kd−1 (sd−1 −1)+kd (sd −2) = [k1 ]s1 · · · [kd ]sd 0
=

X 0

qk1 (s1 −1)+···+kd−1 (sd−1 −1) qkd (sd −2) 1 − qkd + qkd (sd −1) [k1 ]s1 · · · [kd−1 ]sd−1 [kd ]sd

= (1 − q) ζq (s1 , . . . , sd−1 , sd − 1) + ζq (s1 , . . . , sd ) = Sζq (s1 , . . . , sd ). The rest follows easily by induction.



The next corollary answers an implicit question in Kaneko et al. [626]. Corollary 6.22 Let n ∈ N. The specialization of t of gq (s; t) := gq (1; s, t) in 6.8(1) to s − 1 − n is n   X n n gq (s; s − 1 − n) = S ζq (s) = (1 − q)r ζq (s − r), (11) r r=0

where ζq (s − r) := ζq (s − r, 1) in 6.10(4). We observe that one effect of the shifting operator is to bring more poles. Essentially, Sn shifts all the poles of ζq (s), by n to the right on the complex plane.

Analytic Continuation of Multiple Zeta Functions We begin this subsection, by giving a simple observation. Lemma 6.23 For every real number x, we have 4 M! e BM (x) 5 (2π)M

(M ∈ N \ {1}),

(12)

where e Bk (x) denote the periodic Bernoulli polynomials given in 6.10(26). Proof. It follows from 6.10(26) and the fact that ζ (M) 5 ζ (2) = π6 < 2 for M = 2. 2



In Section 2.7, by two different methods, ζ (s1 , . . . , sd ) can be continued to a meromorphic function on Cd with simple poles along Dd , defined by (1). However, neither

534

Zeta and q-Zeta Functions and Associated Series and Integrals

approach is suitable for the present purpose. So, we follow the idea in Section 6.8 to prove a third approach in the rest of this subsection. The same idea will also be used to deal with the q-multiple Zeta functions. To simplify our notation, in the definition of the multiple Zeta function ζ (s1 , . . . , sd ), we replace sd , nd−1 and nd by s, k and n, respectively. Taking f (x) = 1/xs and m = k and ∞ in a Euler-Maclaurin summation formula 2.7(22), we have, for <(s) > 1, ∞

Z ∞ k X X 1 1 1 X 1 = − = f (x) dx − f (k) ns ns ns 2 n=1

n>k

−

n=1

M X r=1

=

k

(−1)M+1 Br+1 (r) f (k) − (r + 1)! (M + 1)!

1 (s − 1) ks−1

Z∞

e BM+1 (x) f (M+1) (x) dx

k

Z∞ e M X 1 Br+1 (s)r (s)M+1 BM+1 (x) − s+ − dx, s+r 2k (r + 1)! k (M + 1)! xs+M+1 r=1

k

where we have used the fact that B2k+1 = 0 (k ∈ N). In view of the observation X

ζ (s1 , . . . , sd ) =

0
s1

k1 k2

s2

X 1 1 , sd−1 ns · · · kd−1 n>k

we have Theorem 6.24 For all (s1 , . . . , sd ) ∈ Cd \ Dd and M > 1 + |σd | + |σd−1 |, we have ζ (s1 , . . . , sd ) =

M+1 X r=0

(sd )M+1 − (M + 1)!

Br (sd )r−1 ζ (s1 , . . . , sd−1 + sd + r − 1) r! X

0
1 s1 s2 k1 k2 · · · kd−1 sd−1

Z∞ e BM+1 (x) dx, xsd +M+1

(13)

kd−1

where (s)−1 := 1/(s − 1). This provides an analytic continuation of ζ (s1 , . . . , sd ) to Cd \ Dd . Proof. We only need to show that the series in (13) converges. Lemma 6.23 implies (if d = 2, then take k0 = 1) ∞ ∞ 1 Z∞ e X X 4 (M + 1)! BM+1 (x) 1 5 , dx sd−1 s +M+1 M+1 M− σ d | |−|σd | kd−1 d−1 x (2π) (M − |σd |) k kd−1 =kd−2 k =k d−1 d−2 d−1 k d−1

which converges (absolutely) whenever M > 1 + |σd | + |σd−1 |. Now, we are ready to prove the main theorem of this section.



q-Extensions of Some Special Functions and Polynomials

535

Theorem 6.25 The multiple q-Zeta function ζq (s1 , . . . , sd ), defined by (3), can be extended to a meromorphic function on Cd with simple poles along Dd (q) given in (4). Further, for all (s1 , . . . , sd ) ∈ Cd \ Dd , lim ζq (s1 , . . . , sd ) = ζ (s1 , . . . , sd ).

(14)

q↑1

Proof. Fix (s1 , . . . , sd ) ∈ Cd , such that σj + · · · + σd < d − j + 1 for all j = 1, . . . , d. When d = 1, this is Theorem 6.16 when a = 1. So, we assume d ∈ N \ {1} and proceed by induction. The key is a recursive formula for ζq (s1 , . . . , sd ), similar to (13) for ζ (s1 , . . . , sd ). To derive this formula, we appeal to the Euler-Maclaurin summation formula 2.7(22) again. Hence, we set F(x) :=

q x(s−1) (1 − qx )s

as in 6.8(17). Then, we have F 0 (x) = (log q) q x(s−1)

s − 1 + qx (1 − q x )s+1

and F 00 (x) = (log q)2 q x(s−1)

s(s + 1) − 3s (1 − q x ) + (1 − q x )2 (1 − q x )s+2

.

In definition (3), for simplicity, we replace sd , kd−1 and kd by s, k and n, respectively. We, now, take M = 1, f (x) = F(x + k − 1), let m → ∞ in 2.7(22) and get ! ∞ X X qn(s−1) s = (1 − q) −F(k) + f (n) [n]sq n=1 n>k ∞  Z Z∞ 1 1 1 e = (1 − q)s  F(x) dx − F(k) − F 0 (k) − B2 (x) f 00 (x) dx 2 12 2 1 k
qk(s−1) 1 qk(s−1) 1 log q qk(s−1) s + qk − 1 q−1 + = − (s − 1) log q [k]s−1 2 [k]sq 12 q − 1 [k]s+1 q q (1 − q)s (log q)2 − 2

Z∞

e B2 (x) q x(s−1)

k

s(s + 1) − 3s (1 − qx ) + (1 − qx )2 (1 − qx )s+2

dx, (15)

because e B2 (x + k − 1) = e B2 (x) by periodicity.

536

Zeta and q-Zeta Functions and Associated Series and Integrals

By the same argument as in Section 6.10, using the incomplete Beta integral given in 6.10(29), we obtain from 6.10(26) the following expression for the last line in (15): X (1 − q)s log q X aν (s) Bqk (s − 1 + δn, −s + ν) (2πin)2

(16)

where δ = 2πi/ log q, a−1 (s) = s(s + 1), a0 (s) = −3s and a1 (s) = 1.



=−

ν=±1, 0

n∈Z\{0}

Repeatedly applying integration by parts on these incomplete Beta integrals, we get for ν = ±1, 0 and positive integer M = 2 Bqk (s − 1 + δn, −s + ν) =

M−1 X

(−1)r−1

r=1

(s + 1 − ν)r−1 qk(s+r−2)
(s − 1 + δn)r 1 − qk s+r−ν

(s + 1 − ν)M−1 + (−1)M−1 B k (s − 2 + M + δn, −s − M + 1 + ν). (s − 1 + δn)M−1 q

(17)

Set Es 0 = (s1 , . . . , sd−2 ), if d = 3 and Es 0 = ∅, if d = 2. Putting (3), (15), (16) and (17) together and applying Theorem 6.21, we get
ζq Es 0, sd−1 , sd =


1
q−1 ζq Es 0, sd−1 + sd − 1 − Sζq Es 0, sd−1 + sd (s − 1) log q 2
s log q 2 S ζq Es 0, sd−1 + sd + 1 + (18) 12 q − 1 X
log q + Sζq Es 0, sd−1 + sd − (Cν + Dν ), 12 ν=±1, 0

where Cν and Dν are contributions from the sum involving Bqk (· · · , −s + ν). We write

T(q, s, n, r) :=

r−1 Y

(2π in + (s − 1 + j) log q)−1

(19)

j=0

to express Cν and Dν explicitly as follows: C−1 =

M−1 X

X T(q, s, n, r)  log q r+1
(s)r+1 S3 ζq Es 0, sd−1 + sd + r + 1 2 q−1 (2πin)

r=1 n∈Z\{0}

and D−1 =

X 0
qk1 (s1 −1)+···+kd−2 (sd−2 −1) s [k1 ]sq1 · · · [kd−2 ]qd−2

∞ X k=kd−2 +1

R(M, q, k, sd−1 , s),

q-Extensions of Some Special Functions and Polynomials

537

where we replace the index kd−1 by k and R(M, q, k, sd−1 , s) := −

qkd−1 (sd−1 −1) s [kd−1 ]qd−1

· (s)M+1

Z∞

X T(q, s, n, M − 1)  log q M+1 q−1 (2π in)2

n∈Z\{0}

e2π inx q x(s−2+M) [x]q−s−M−1 dx; (20)

kd−1 M−1 X

X T(q, s, n, r)  log q r q−1 (2πin)2 r=1 n∈Z\{0}
· (s)r S2 ζq Es 0, sd−1 + sd + r

C0 = 3 log q

and X

D0 = −3 log q

0


log q · q−1 C1 = (log q)2

M

(s)M

Z∞

qk1 (s1 −1)+···+kd−1 (sd−1 −1) X T(q, s, n, M − 1) s [k1 ]sq1 · · · [kd−1 ]qd−1 n∈Z\{0} (2π in)2

e2π inx q x(s−2+M) [x]−s−M dx; q

kd−1

M−1 X

X T(q, s, n, r)  log q r−1
(s)r−1 Sζq Es0, sd−1 + sd + r − 1 2 q−1 (2πin)

r=1 n∈Z\{0}

and X

D1 = −(log q)2

0
 ·

log q q−1

M−1

(s)M−1

Z∞

qk1 (s1 −1)+···+kd−1 (sd−1 −1) X T(q, s, n, M−1) s [k1 ]sq1 · · · [kd−1 ]qd−1 n∈Z\{0} (2πin)2

dx. e2π inx q x(s−2+M) [x]−s−M+1 q

kd−1

The next crucial step is to control the summation over kd−1 and show that they converge uniformly with respect to q. When 0 < q 5 1/2, this is clear. The only nontrivial part is when q ↑ 1. So, we assume 1/2 < q < 1. Note that lim T(q, s, n, r) = q↑1

1 (2πin)r

and

lim q↑1

log q = 1. q−1

Lemma 6.26 Let sd = s = σ + i τ and   max 1/2, e(6−2π )/τ (τ > 0), q0 := 1/2 (τ 5 0).

538

Zeta and q-Zeta Functions and Associated Series and Integrals

Then, for all q0 < q < 1 and k ∈ N, we have log q 1 q − 1 < 2 and |T(q, s, n, r)| < (6n)r . Proof. Let f (q) = 2(1 − q) + log q. Then, f 0 (q) = −2 + 1/q < 0 whenever q > q0 . So, f is strictly decreasing on q0 < q. Then, we have f (q) > f (1) = 0 whenever q0 < q < 1. This implies that 2(1 − q) > − log q from which log q/(q − 1) < 2.  To bound T(q, s, n, r), we consider each of its factors in definition (19). For each 0 5 j < r, we have |2π in + (s − 1 + j) log q|2 = {(σ − 1 + j) log q}2 + (2πn + τ log q)2 = (2πn + τ log q)2 , which is independent of j ( j = 0, 1, . . . , r − 1). If τ 5 0, then clearly |2πin + (s − 1 + j) log q| > 6n. If τ > 0, then it follows from q > e(6−2π )/τ that 2πn + τ log q > 2πn + 6 − 2π = 6(n − 1) + 6 = 6n as desired. Next, we want to bound the integral terms in D−1 . Let |σd | < N and |σd−1 | < N 0 for some positive integers N and N 0 . Fix an arbitrary x > k = kd−1 and a positive integer M > 16 + 2N + 6

d−1 X


Nj + 1 ,

j=1

where we recall |<(sj )| < Nj ∈ N ( j = 1, 2, . . . , d) and so N = Nd and N 0 = Nd−1 . Then, we have 0 0 q−k(M/6−N −1) qk(s −1) q x(s−2+M) [x]−s−M−1 < g(q), q where s0 = sd−1 , and, for convenience, g(q) is given by g(q) = q x(M−N−2)−

kM 6

[x]−M+N−1 . q

(21)

Lemma 6.27 Let 1/2 < q < 1. Then, g(q) is increasing as a function of q, so that g(q) 5 lim g(q) = q↑1

1 xM−N+1

.

Proof. Taking the logarithmic derivative of g(q), we have g0 (q) h(q) = , g(q) q(1 − q) (1 − qx )

q-Extensions of Some Special Functions and Polynomials

539

where h(q) := (1 − q) 1 − qx




 x(M − N − 2) −

kM 6



n o + (M − N + 1) x qx (1 − q) − q + q x+1 . Then, we get  n o kM h0 (q) = (x + 1) qx − x q x−1 − 1 x(M − N − 2) − 6 n o + (M − N + 1) x2 q x−1 − x(x + 1) qx − 1 + (x + 1) qx . Clearly, we have h0 (1) = 0. Moreover, it is easily seen that n o q2−x h00 (q) = (kM/6 + 3x) x2 (1 − q) − x(1 + q) + qx(x + 1)(M − N + 1) > qx(x + 1)(M − N + 1) − x(1 + q) (kM/6 + 3x)   1+q = qx2 M − N + 1 − (3 + M/6) q > qx2 (M/2 − N − 8) > 0, where we used the fact that if q > 1/2, then (1 + q)/q < 3. This implies that h0 (q) is increasing, by recalling h0 (1) = 0, so that h0 (q) < 0, for all 1/2 < q < 1. It follows that h(q) is decreasing. Since h(1) = 0, we find that h(q) > 0 for all such q. Therefore, g0 (q) > 0, and so g(q) is increasing for all 1/2 < q < 1. This completes the proof of the lemma.  We, now, can bound the innermost sum of D−1 . It follows from Lemma 6.17, Lemma 6.26 and Lemma 6.27 that (if d = 2, then take k0 = 1) ∞ X k=1+kd−2

R(M, q, k, s0, s) <

∞ X

(2k)N

k=1+kd−2

0

2 ζ (M + 1) M+1 2 4 π 2 6M−1

k(M/6−N 0 −1)

· (N)M+1 q

Z∞

dx xM−N+1

k

  (M + 1)! M + N < M−N M+1

∞ X k=1+kd−2 ∞ X

< (M + 1)! (2M)M+1

k=1+kd−2 0

0

since 2N +M+2 ζ (M + 1) < 4 π 2 6M−1 and M − N > 2.

qk(M/6−N −1) 0 kM−N−N 0

qk(M/6−N −1),

540

Zeta and q-Zeta Functions and Associated Series and Integrals

Therefore, by Lemma 6.17, we have 

d−2 Y

X

|D−1 | < (2M)2M+2

 0
 kj j qkj (−Nj −1)  qkd−2 (M/6−Nd−1 −1), N

j=1

which converges as proven in Theorem 6.18. Exactly the same argument can apply to the integral terms in D0 and D1 . These convergence results imply two things. First, we can show by induction on d that (18) gives rise to an analytic continuation of ζq (s1 , . . . , sd ) as a meromorphic function on Cd \ Dd (q). Second, also by induction on d, we, now, can conclude that it is legitimate to take the limit q ↑ 1 inside the sums of Cν and Dν to get (note that, for any n ∈ N,   lim Sn ζq (Es) = ζ (Es) Es ∈ Cd−1 \ Dd−1 (q) q↑1


lim ζq Es 0, sd−1 , sd = q↑1

+


1
1 ζ Es 0, sd−1 + sd − 1 − ζ Es 0, sd−1 + sd s−1 2

X X

M−1 s 1 (s)r+1 ζ Es0, sd−1 + sd + r + 1 ζ Es0, sd−1 + sd + 1 − r+2 12 (2πin) r=1 n∈Z\{0}

X

+

0
=

M+1 X r=0

1 sd−1 s1 k1 · · · kd−1

X n∈Z\{0}

(s)M+1 (2πin)M+1

Z∞

e2π inx x−s−M−1 dx

kd−1

Br (sd )r−1 ζ (s1 , . . . , sd−1 + sd + r − 1) r!

1 − (M + 1)!

X 0
1 sd−1 s1 k1 · · · kd−1

Z∞

e BM+1 (x)

(s)M+1 dx, xs+M+1

kd−1

by 6.10(26) and its specialization with x = 1: X n∈Z\{0}

e Br+2 1 Br+2 (1) =− . =− r+2 (r + 2)! (r + 2)! (2πin)

The main theorem, Theorem 6.25, now mostly follows from Theorem 6.24. The 2π i poles at sd = m − log q n are given, by the first term in the formula (18) when m = 1 and n = 0 and by the terms T(q, sd , n, r), as defined in (19), if m 5 1 and n 6= 0. The locations of the other poles are obtained by induction, using those poles of the q-generalized Zeta function presented in Theorem 6.13. This completes the proof of the main theorem.

q-Extensions of Some Special Functions and Polynomials

541

Special Values of ζq (s1 , s2 ) For integers n1 , . . . , nd , we set ζq (n1 , . . . , nd ) = lim · · · lim ζq (s1 , . . . , sd ) sd →nd

s1 →n1

and ζqR (n1 , . . . , nd ) = lim · · · lim ζq (s1 , . . . , sd ) sd →nd

s1 →n1

if the limits exist. Interesting phenomena occur already in the case d = 2, and these should be generalized to arbitrary depth. By Theorem 6.20, we get ζq (s1 , s2 ) = (1 − q)s1 +s2

  ∞  X s1 + r1 − 1 s2 + r2 − 1 r1 r2

r1 , r2 =0

q2s2 +2r2 +s1 +r1 −3

·



1 − qs2 +r2 −1 1 − qs2 +r2 +s1 +r1 −2 ( q2s2 +s1 − 3 s1 q2s2 +s1 − 2 s1 +s2

+

= (1− q) s − 1 s +s − 2 s 1− q 2 1− q 2 1 1− q 2 − 1 1− qs2 +s1 − 1 +

s1 s2 q2s2 +s1 s2 q2s2 +s1 −1
+
(1 − qs2 ) 1 − qs2 +s1 −1 (1 − qs2 ) 1 − qs2 +s1

) s2 (s2 + 1) q2s2 +s1 +1 s1 (s1 + 1) q2s2 +s1 −1

+

+ ··· . + 2 1 − qs2 −1 1 − qs2 +s1 2 1 − qs2 +1 1 − qs2 +s1 Clearly, we have ζq (0, 0) = lim lim ζq (s1 , s2 ) = s1 →0 s2 →0

ζqR (0, 0) = =

1


q2 − 1

(q − 1)

−

3 1 ; + 2(q − 1) log q log2 q

lim lim ζq (s1 , s2 )

s2 →0 s1 →0

1 1 q
− + . q2 − 1 (q − 1) (q − 1) log q 2(q − 1) log q

We readily find that lim ζq (0, 0) = q↑1

1 3

and

lim ζqR (0, 0) = q↑1

5 , 12

which is consistent with those limiting values in 2.7(18).

(22)

542

Zeta and q-Zeta Functions and Associated Series and Integrals

Problems 1. Show that X

lim (1 − q)s log q q↑1

n∈Z\{0}

e−2πina s Bqa (s + δn − 1, −s) = 0. (2π in)2 (See the last part of Section 6.10)

2. Show the limiting values in 6.11(22): 1 3

lim ζq (0, 0) = q↑1

and

lim ζqR (0, 0) = q↑1

5 . 12

We may use the following asymptotic formulas:   1 t t2 1 = − + + O t3 (t → 0); t+2 2 4 8   1 1 1 t t2 = + − + + O t3 (t → 0); log(1 + t) t 2 12 24   1 1 1 1 + 0 · t + O t2 = 2+ + (t → 0). 2 t 12 log (1 + t) t 3. For notations, here, see Problem 48 in Chapter 2. The q-analogues of MZVs, MZSVs and the multiple polylogarithm with equality are defined, respectively, as follows: ζq [k1 , . . . , kr ] : =

X n1 >···>nr >0

X

ζq∗ [k1 , . . . , kr ] : =

n1 =···=nr =1

qn1 (k1 −1)+···+nr (kr −1) , [n1 ]k1 · · · [nr ]kr qn1 (k1 −1)+···+nr (kr −1) [n1 ]k1 · · · [nr ]kr

and X

Li∗k1 ,...,kr [t] :=

n1 =···=nr =1

tn1 . [n1 ]k1 · · · [nr ]kr

Prove the following results: (a) Sum Formula. For r < k (r, k ∈ N), X k∈I0 (k, r)

  r−1   1 k−1 X r−1 ζ [k] = (k − 1 − `) (1 − q)` ζq [k − `]. k−1 r−1 ` ∗

`=0

(b) Cyclic Sum Formula. For (k1 , . . . , kr ) ∈ I0 (k, r), r kX i −2 X

ζq∗ [ki − j, ki+1 , . . . , kr , k1 , . . . , ki−1 , j + 1]

i=1 j=0

=

r X `=0

  r (k − `) (1 − q)` ζq [k − ` + 1], `

where the empty sum is to be interpreted as nil.

q-Extensions of Some Special Functions and Polynomials

543

(c) Generating Function of Multiple Polylogarithms.  X  X k>r>0

 

k∈I0 (k, r)



1 = 1−u−v

Li∗k [t] uk−r−1 vr−1 

Zt

(s; q)∞ 2 81 (a, b; aq; q, s) dq s, (bs; q)∞

0

where 2 81 is the q-hypergeometric series (see 6.3(28)), the integral is the Jackson q-integral given in 6.2(10), q−a−1 =

1 1 − (1 − q)(u + v)

and

b=

1 − (1 − q)u . 1 − (1 − q)(u + v)

(see [873, Theorems 1 and 2 and Corollary 3]) a. For m ∈ N0 , m   ∞ n(k−m) X X m q = (1 − q)` ζq [k − ` + 1]. ` [n]k+1 `=0

n=1

(See [873, p. 3033]) 4. Continuing Problem 4, denote the q-analogue of MZVs by MqZVs, for short. Define a generating function of MqZVs as follows:

80 [x, y, z] :=

  ∞   X

X

ζq [k]

  k,r,s=0 

k |k|=k, dep k=r, ht k=s

   

xk−r−s yr−s zs−1 .

  

Then, 1 + (z − xy) 80 = exp

(∞ X n=2

) ∞ X
(q − 1)m m+n m+n m+n m+n ζq [n] x +y −α −β , m+n m=0

where α m+n + β m+n is a polynomial in x, y and z, determined by α + β = x + y + (q − 1)(z − xy)

and

α β = z. (See [875, Theorem 1])

 5. If r ∈ N and z ∈ C \ q−m [m]q | m ∈ N , then show that X k1 >···>kr >0

r ∞ X qk1 X 1 qrm
. = k r j [k1 ]q [m]q [m]q − z qm [kj ]q − z q j=1 m=1

(See [169, Theorem 2])

544

Zeta and q-Zeta Functions and Associated Series and Integrals

6. Let n, m and s1 , s2 , . . . , sm be nonnegative integers. Define the multiply-nested sums m Y

X

Zn [s1 , . . . , sm ] :=

−s

qkj [kj ]q j ,

n=k1 =···=km =1 j=1

(−1)k1 +1 qk1 (k1 +1)/2

X

An [s1 , . . . , sm ] :=

n=k1 =···=km =1



n k1

 Y m

q(sj −1)kj [kj ]q j , −s

q j=1

with the understanding that Z0 [s1 , . . . , sm ] = A0 [s1 , . . . , sm ] = 0 and, with empty argument lists, Zn [ ] = An [ ] = 1, if n ∈ N and m = 0. The sums are over all integers kj , satisfying the indicated inequalities.  For convenience, Catm j=1 sj denotes the concatenated argument sequence s1 , . . . , sm and {s}m = Catm j=1 {s} (m ∈ N0 ) consecutive copies of the argument s. For n, r and a1 , b1 , . . . , ar , br ∈ N, show that    r−1 n o o r n Zn Cat {1}aj −1 , bj + 1 , {1}ar −1 , br = An a1 , {1}b1 −1 , Cat aj + 1, {1}bj −1 . j=1

j=2

(See [166, Theorem 1]) 7. If n ∈ N and s > 1, then show that n ζ {s} 

n



=

n X

(−1)

k+1

 k−1  h iX k−1 n−k ζ {s} (1 − q)j ζ [ks − j]. j j=0

k=1

(See [167, Theorem 1]) 8. For s, t ∈ N \ {1}, show that ζ [s] ζ [t] =

s−1 s−1−a X a + t − 1 t − 1 X

t−1

a=0 b=0

+

(1 − q)b ζ [t + a, s − a − b]

t−1 t−1−a X X a + s − 1 s − 1 a=0 b=0

−

b

min(s,t) X j=1

s−1

b

(1 − q)b ζ [s + a, t − a − b]

(s + t − j − 1)! (1 − q)j ϕ[s + t − j], (s − j)! (t − j)! ( j − 1)!

where, for convenience, the sum ϕ[s] is given by ϕ[s] :=

∞ ∞ X (n − 1) q(s−1)n X n q(s−1)n = − ζ [s]. s [n]q [n]sq n=1

n=1

(See [168, Theorem 2.1]) 9. Let σj (n) denote the sum of the jth powers of the divisors of n, given by σj (n) :=

X d|n

dj

and

Sj (q) :=

∞ X n=1

σj (n) qn (q ∈ C; |q| < 1).

q-Extensions of Some Special Functions and Polynomials

545

These generating functions Sj (q) are known to have the following expansions as Lambert series: Sj (q) =

∞ X n j qn . 1 − qn n=1

The simplest case is j = 0 when σ0 (n) = d(n), the divisor function counting the numbers of divisors of n. It was shown by Uchimura [1171] that ∞ X

∞ X

d(n) qn =

n=1

∞  Y

n qn

n=1

 1 − qj .

j=n+1

Uchimura [1173] studied the following more general expressions: Uk (q) :=

∞ X

nk qn

n=1

∞   Y 1 − qj

(k ∈ N).

j=n+1

Show that Uk (q) =

X

k! n1 ! · · · nk !



S0 (q) 1!

n1 

S1 (q) 2!

n2

 ···

Sk−1 (q) k!

nk

,

where the summation extends over all integers n1 , . . . , nk = 0

with

n1 + 2 n2 + · · · + k nk = k. (See [1173])

10. Continued from Problem 9, show that (a) for k ∈ N, we have Sk−1 (q) =

    X (−1)n1 +···+nk −1 Uk (q) nk U1 (q) n1 ··· , (n1 + · · · + nk − 1) ! n1 ! · · · nk ! 1! k!

where the summation extends over all integers n1 , . . . , nk = 0

with

n1 + 2 n2 + · · · + k nk = k; (See [383, Theorem 1])

(b) for k ∈ N, we have    k−i  k X X k−1 s( j + i, i)  Ui (q), =  j + i − 1 ( j + i)!  (q; q)m (1 − qm )k i=1 j=0

m+1 ∞ X (−1)m−1 q( 2 )

m=1

where the s(m, n) are the Stirling numbers of the first kind (see Section 1.5). (See [383, Theorem 3]) 11. Let m, n ∈ N, τ = m − n + 1, a, b, z, q be variables and ( ) a − bq a − bq2 a − bqn A= , , ..., . c − zq c − zq2 c − zqn

546

Zeta and q-Zeta Functions and Associated Series and Integrals

Show that X 15i1 5i2 5···5ir 5n

a − bqi1 a − bqi2 a − bqir · · · c − zqi1 c − zqi2 c − zqir



m i+1 n   X n (−1)i−1 q( 2 )−ni 1 − qi a − b qi cn (zq/c; q)n = .
τ +1 i q (q; q)n (az − bc)n−1 c − z qi i=1

(See [464, Proposition 2.1]) 12. Show that n   X n k=1

k

(−1)k f (k) =

(−1)n 2πi

Z C

n! f (z) dz, z(z − 1) · · · (z − n)

where C is a positively-oriented (counter-clockwise) closed contour in the complex z-plane, which encloses the poles at z = 1, 2, . . . , n and no other poles of the integrand. By extending the contour of integration and taking the extra negative residues into account, one can get a number of identities, provided that the integral goes to zero, which, incidentally, it does for a large class of functions f (z). (See [916, p. 552]) 13. Let   d 0q (z) . ψq (z) := dz 0q (z) Show that (    ) ∞ X B2k 1 − qz ln q d 2k−1 qz+t ln q
− ψq (z) ∼ ln + z+t −z 1−q (2k)! dt 1−q 2 q −1 t=0 k=1     ∞ 2k X B2k
ln q ln q 1 − qz
− qz P2k−2 qz , + = ln z −z 1−q (2k)! (1 − q) 2 q −1 k=1 

where Pn (z) is a polynomial of degree n, satisfying the following recurrence relation:   Pn (z) = z − z2 P0n−1 (z) + (nz + 1) Pn−1 (z) (n ∈ N),

P0 = 1,

where the principal branch of the logarithm is taken. (See [838, Theorem 1]) 14. Moak [836, Equation (1.1)] defines 0q (q > 1) as follows: q(2) q−1 ; q−1 x

0q (x) =




(q − 1)1−x

∞
q−x ; q−1 ∞

.

Show that this 0q (q > 1) also satisfies the functional equation 0q (x + 1) = [x]q 0q (x).

q-Extensions of Some Special Functions and Polynomials

547

15. One of the most interesting q-analogs of harmonic numbers Hn seems to be Hn (q), given by Hn (q) =

n X k=1

qk . 1 − qk

Show that   n X (−1)k−1 k(k+1)/2 n q = Hn (q). k q 1 − qk k=1

(See [1185]; see also [52]) 16. Let p be an odd prime and let n ∈ N and r ∈ Z. If r ∈ N, then show that np X j=1 ( j,p)=1

  ∞   X −r 1 = − Lp r + k, ω1−k−r (pn)k r j k k=1

and 2n X j=1 ( j,2)=1

 ∞  X 1 −r = − L2 (r + k, 1) (2n)k , jr k k=1

where Lp (s, χ ) ( p a prime) denotes the p-adic L-function attached to a Dirichlet character χ and ω the Teichmu¨ ller character. (See [1210, Theorem 1]) 17. The q-logarithm function is defined by lq (x) =

∞ X (−1)n−1 (x − 1)n [n]q

(|x − 1| < 1),

n=1

where q = (q1 , . . . , qr )

(0 < q1 , . . . , qr < 1)

and

[n]q = [n]q1 · · · [n]qr .

Let 0 < qj < 1 ( j = 1, . . . , r). The q-Mahler measure mq ( f ) of a rational function f (x1 , . . . , xn ) ∈ C (x1 , . . . , xn ) is defined as follows: mq ( f ) = <

 1 Z 

0

Z1 ··· 0

     lq f e2π iθ1 , . . . , e2π iθn dθ1 · · · dθn . 

Show that   x+1 4 X an mq az +1 = 2 y+1 π [n]q n2

(0 < a 5 1).

n=1 (n odd)

(See Gon et al. [493, Theorem 1])

548

Zeta and q-Zeta Functions and Associated Series and Integrals

18. Subject to suitable convergence conditions, if we, suppose that cn =

∞ X

am+n bm ,

m=0

then show that ∞ X

bm

∞ X n=−∞

m=0

∞ X

an =

cn .

n=−∞

(See Andrews [46, Transformation Lemma]) 19. If h(z) is given by   h(z) = −zq2 ; q4

∞ X ∞

n=0

! 2 2n ∞   X q4n z 2 −n 2n z −1 2 4
+ q −z q ; q , n −zq2 ; q4 n n=1

then show that


  q2 ; q2 ∞ −zq; q2 ∞ −z−1 q; q2 ∞


. h(z) + zq h z q2 = −q2 ; q2 ∞ q; q5 ∞ q4 ; q5 ∞ (See Andrews [46, Theorem 2]) 20. The coefficients Bn (α, θ, q) are defined by ∞ X

∞ Q

Bn (α, θ, q) xn =

n=−∞

1 + 2xqn cos θ + x2 q2n

n=1 ∞ Q


.

1 + α qn x eiθ




n=1

Show that ∞ X

q−n Bn+m (α, θ, q) Bn (β, θ, q) =

n=−∞


m (αq)∞ (βq)∞ (1/α)m −α q eiθ {(q)∞ }2 ∞  m  n X q · (1/β)n α β q e2iθ . α n n=−∞ (See Bhargava et al. [132, Theorem])

21. A q-analogue of the usual logarithm is defined by `q (x) =

∞ X (−1)n−1 (x − 1)n . [n]q n=1

Recall the quantum dilogarithm given by (see Kirillov [667]): Li2,q (x) =

∞ X n=1

xn . n [n]q

q-Extensions of Some Special Functions and Polynomials

549

Kurokawa and Wakayama [719] defined the q-function Fq (x) by Fq (x) :=

∞  q−1 Y 1 − q−n e2π ix . n=1

Show that π

Z2 0

  1 `q 1 − a e2ix dx = − Li2,q (−a) − Li2,q (a) 2i    log a 1 F + q 2π i 2 1   (0 < a 5 1).  = log  log a 2i F q

2π i

(See Kurokawa and Wakayama [719, Theorem 1.2]) 22. Show that Z1

ζ (s, x) dq x =

 ∞  X 1 −s ζ (s + k) + [1 − s]q k [k + 1]q

(<(s) < 1)

k=0

0

and Z1

ζ (s, 1 − x) dq x =

∞ X

(−1)k

k=0

0

  −s ζ (s + k) k [k + 1]q

(s ∈ C \ {1}),

where ζ (s, x) and ζ (s) denote the generalized (or Hurwitz) zeta function and the Riemann zeta function, and the Jackson integral of a suitable function f is defined by Z1

f (x) dq x = (q − 1)

∞ X


f q−n q−n

(q > 1).

n=1

0

(See Kurokawa et al. [715, Theorem 2]) 23. For n ∈ N, define the Ap´ery number A(n) by A(n) :=

   n  X n+j 2 n 2 j=0

j

j

and suppose that the integers a(n) are given by the following generating function: ∞ X n=1

a(n) qn = q

∞  Y n=1

1 − q2n

4 

1 − q4n

4

.

550

Zeta and q-Zeta Functions and Associated Series and Integrals

Show that, for every odd prime p,  A

p−1 2

 ≡ a(p)

(mod p).

(Beukers [127]; see also Ahlgren and Ono [14, Eq. (1.6)]) 24. Derive the q-Pfaff-Saalschu¨ tz formula:  c  ;q n b  n . q, q = a 3 82  c ;q (c ; q)n c, abc−1 q1−n ; ab n a, b, q−n ;





c

;q

See, e.g., Wang [1206, Theorem 4.1]; see also Srivastava ([1070] and
[1077]) for more general results 25. Show that n X

4(n−k)

q

k=1



2  1 + q2 − 2 qk+1 1 − qk n+1 2
. = 2 q (1 − q)2 1 − q2

(Zhao and Feng [1259, Eq. (2.1)]) 26. The q-analogue of the Psi function ψ(x) (Section 1.3) is defined as the logarithmic derivative of the q-Gamma function, that is, ψq (x) =

0q0 (x) 0q (x)

,

so that we find from 6.2(2) that ψq (x) = −Log(1 − q) + Log q

= −Log(1 − q) +

Log q 1−q

∞ X n=0 ∞ X

qn+x 1 − qn+x qnx 1 − qn

n=1

(x > 0),

which implies that Log q ψq (x) = −Log(1 − q) + + 1−q

Zq 0

tx−1 dq t. 1−t

For q ∈ (0, 1), show that  2 ψq,m (x) ψq,n (x) = ψq, m+n (x) 2

  m+n x > 0; m, n, ∈N , 2

(n)

where ψq,n (x) := ψq (x). (Brahim [170]; see also [699])

q-Extensions of Some Special Functions and Polynomials

551

27. For a suitably bounded sequence {n }n∈N0 of essentially arbitrary complex numbers, show that the following multiple q-summation formula holds true: (γ qn ; q)k+k1 +···+kr (β1 ; q)k1 · · · (βr ; q)kr (q; q)k1 · · · (q; q)kr (γ ; q)k+k1 +···+kr (β1 q; q)k1 · · · (βr q; q)kr k=0 k1 ,...,kr =0  r ∞ (q; q)n (γ / (β1 · · · βr ) ; q)n X (γ qn / (β1 · · · βr ) ; q)k = (β1 · · · βr )n ,  (k) (γ ; q)n (β1 q; q)n · · · (βr q; q)n (γ / (β1 · · · βr ) ; q)k

∞ X

n X

q−n ; q




 (k)

k1 · · ·

q−n ; q




kr

k=0

provided that the series involved are absolutely convergent. (Lin and Srivastava [766, p. 319]; see also Problem 62 of Chapter 1 for the limit case when q → 1) (α) 28. For the basic (or q-) Genocchi polynomials Gn;q (x) and the basic (or q-) Genocchi numbers (α) (α) e G and G (see, e.g., Section 1.7), defined for q, α ∈ C (|q| < 1) by n;q

n;q

(2z)α

∞ X ([α]q )n n=0

[n]q !

(−1)n qn+x ez[n+x]q =

∞ X

(α)

Gn;q (x)

n=0

zn n!

and   (α) (α) α e Gn;q := 2n Gn;q , 2 respectively, so that   1 (1) Gn;q (x) := Gn;q (x) and e Gn;q := 2n Gn;q 2

(n ∈ N0 ),

derive the following limit formulas: n o (α) lim Gn;q (x) = G(α) n (x)

q→1

and

n o (α) lim Gn;q = G(α) n .

q→1

(cf., e.g., Srivastava [1091]; see also the many references relevant to this subject, which are cited therein, as well as in this book) Remark 5 For basic (or q-) extensions of the Apostol-Genocchi polynomials Gn (x; λ) (α) and the Apostol-Genocchi polynomials Gn (x; λ) of (real or complex) order α, defined by 1.8(56), the interested reader may refer to a number of recent works by (among others) Luo [786].

552

Zeta and q-Zeta Functions and Associated Series and Integrals

 29. For suitably bounded sequences {n }n∈N0 and 3m,n m,n∈N , show that 0

` Q ∞ X n=0




cj ; q

j=1 m Q

[n/N] tn X

n

n
(q; q)n

dj ; q

n

j=1

q−n ; q

m Q


Nk

` Q

k=0

q1−n /dj ; q




j=1

q1−n /cj ; q


Nk

j=1




·q

1−(`−m−1)n Nk

3n,k zk (q; q)k ` n Q

∞ X

=

q− 2 (`−m−1)Nk(Nk+1) 1

n,k=0

j=1 m n Q j=1

z (−1)`−m+1 t (q; q)k 

·

Nk


o

cNk j · cj ; q

n

tn  3
o n+Nk n+Nk,k (q; q)n

djNk · dj ; q

n

N
k (N ∈ N).

Also deduce its special case, involving the basic (or q-) hypergeometric function r 8s , defined by 6.3(28). (See Srivastava [1066, p. 334, Eq. (3.9)]) (α) 30. For q, α ∈ C (|q| < 1), let the q-Bernoulli polynomials Bn;q (x) of order α in qx and the (α)

q-Euler polynomials En;q (x) of order α in qx be defined by means of the following generating functions: (−t)α

∞ X ([α]q )n n=0

[n]q !

qn+x e[n+x]q t =

∞ X

(α)

Bn;q (x)

n=0

tn n!

and 2α

∞ X ([α]q )n n=0

[n]q !

(−1)n qn+x e[n+x]q t =

∞ X n=0

(α)

En;q (x)

tn , n!

respectively. Show that each of the following relationships holds true:  n    X n 1 (k−α)x (α) (α) q Bk;q (y) + qn−k−x−α+2 Bk;q;x (y) k 2 k=0  1 (α−1) + kqn−k−x−α+2 Bk−1;q;x (y) En−k;q (x) (n ∈ N0 ; α ∈ C), 2     n X 1 n (α) (α−1) (α) n−k−x−α+1 En;q (x + y) = q 2Ek+1;q;x (y) − Ek+1;q;x (y) k+1 k k=0  2α qy (qn+1 − 1) (α) Bn+1;q (x) − q(k+1)x Ek+1;q (y) Bn−k;q (x) + (n + 1)(−q; q)α (α)

Bn;q (x + y) =

(n ∈ N0 ; α ∈ C)

q-Extensions of Some Special Functions and Polynomials

553

and (1)

Bn;q (x) := Bn;q (x) =

 n    X n 1 (k−1)x q Bk;q + qn−k−x+1 Bk;q;x (0) k 2 k=0  1 + kqn−k (1 − q x−1 )k−1 En−k;q (x), 2

(α)

(α)

where the polynomials Bn;q;y (x) and En;q;y (x) in qx are as follows: (α)

Bn;q;y (x + 1) =

n   X n

k

k=0

(α)

q(k−α+1)y Bk;q (x)

and (α)

En;q;y (x + 1) =

n   X n (k+1)y (α) q Ek;q (x), k k=0

which yield the following obvious relationships: (α)

(α)

Bn;q;1 (x) = Bn;q (x)

and

(α)

(α)

En;q;1 (x) = En;q (x),

(2.15)

respectively. We also assume (as usual) that Bn;q;y (x) := B1n;q;y (x)

and

1 (x). En;q;y (x) := En;q;y

(See Luo and Srivastava [790, p. 249, Theorem 1; p. 251, Theorem 2]; see also Srivastava and Pint´er [1105, p. 379, Theorem 1; p. 380, Theorem 2], Cheon [254, p. 368, Theorem 3] and Sections 1.7 and 6.6 to 6.9) 31. Making use of the definitions and notations in Problem 30 above, show that (α)

(α)

(α−1)

qα−1 Bn;q (x + 1) − Bn;q (x) = nBn−1;q (x) (n ∈ N \ {1}), n   X n (α) (α) Bn;q (x + y) = B (x)q(k−α+1)y [y]n−k q , k k;q k=0 (α) (α−1) α−1 (α) q En;q (x + 1) + En;q (x) = 2En;q (x)

(n ∈ N \ {1})

and (α) En;q (x + y) = α−1

q

n   X n

k

(α)

Ek;q (x)q(k+1)y [y]n−k q ,

k=0 (α) (α) (α−1) Bn;q;y (x + 1) − Bn;q;y (x) = nBn−1;q;y (x)

(n ∈ N \ {1})

and (α)

(α)

(α−1)

qα−1 En;q;y (x + 1) + En;q;y (x) = 2En;q;y (x)

(n ∈ N0 ).

(See Luo and Srivastava [790, pp. 245–248, Lemmas 1 to 3])

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7 Miscellaneous Results Over one and a half centuries ago, von Staudt and Clausen proved independently (and published almost simultaneously) a useful theorem for the calculation of the higherorder Bernoulli numbers, which has recently been generalized and proven in many different ways in various investigations dealing with evaluations of the Bernoulli and Euler polynomials at rational arguments. Closed-form summations of several trigonometric series have also been of recent concern by various authors. In addition to presenting some of these developments in this chapter, we establish several integrals involving a certain periodic function and various other families of definite integrals involving logarithms of the Gamma function, by introducing some not-too-familiar mathematical constants that stem from the use of the Euler-Maclaurin summation formula 1.3(68).

7.1 A Set of Useful Mathematical Constants There are some classes of mathematical constants involved naturally in the Gamma and multiple Gamma functions. Here, we introduce those known mathematical constants associated with the Gamma and multiple Gamma functions and show how they are involved, if possible (see Choi [265]).

Euler-Mascheroni Constant γ In addition to the brief history of γ and its various integral representations in Chapter 1, it is remarked that the Euler-Mascheroni constant γ is the third important mathematical constant next to π and e, whose transcendence were shown by Ferdinand Lindemann in 1882 and Charles Hermite in 1873, respectively. The mathematical constants π , e and γ are often referred to as the holy trinity. Very recently, in a tantalizing blend of history and mathematics, Havil takes us on a journey through logarithms and the harmonic series, the two defining elements of γ as in 1.1(3), toward the first account of γ ’s place in mathematics in his book [543]. Finch [449, pp. 35–40] recorded a fairly comprehensive bibliography of as many as 136 references on the γ . We can show that the infinite product ∞  Y z z 1 + e− n n

n=1

Zeta and q-Zeta Functions and Associated Series and Integrals. DOI: 10.1016/B978-0-12-385218-2.00007-4 c 2012 Elsevier Inc. All rights reserved.

556

Zeta and q-Zeta Functions and Associated Series and Integrals

converges in the finite complex plane C to an entire function that has simple zeros at z = −1, −2, −3, . . . , this argument yields that ∞  Y z −1 z 1+ en n

n=1

converges on every compact subset in C \ {−1, −2, −3, . . . } to a function with simple poles at z = −1, −2, . . . . Using this fact, Weierstrass defined the Gamma function, 0(z), is a meromorphic function on C with simple poles at z = 0, −1, −2, . . . , given by (see 1.1(2)) 0(z) =

∞ e−c z Y  z −1 z 1+ en , z n

(1)

n=1

where c is a constant chosen so that 0(1) = 1. Indeed, it is easy to show that the chosen constant c in (1) corresponds to the Euler-Mascheroni constant γ in 1.1(3).

Series Representations for γ Here, we summarize some known series representations for γ (some of which are seen already recorded in Chapter 3, yet, for completeness, rewritten here) and point out that one of those series representations seems incorrectly recorded (see [264, 284, 543]). We start by recalling a well-known series representation for γ : γ=

∞ X

(−1)k

k=2

ζ (k) , k

(2)

where the Riemann Zeta function ζ (s) is defined by 2.3(1). γ = 1−

∞ X ζ (k) − 1 . k

(3)

k=2

γ = 1 − log 2 + γ = 1−

∞ X

k=2 ∞ X

1 log 2 − 2

γ = log 2 − γ = log 2 −

(−1)k

k=1

ζ (k) − 1 . k

ζ (2k + 1) − 1 . 2k + 1

∞ X ζ (2k + 1) − 1 . k+1 k=1 ∞ X k=1

ζ (2k + 1) . (2k + 1) 22k

(4)

(5)

(6)

(7)

∞

γ = 1 − log

3 X ζ (2k + 1) − 1 − . r2 (2k + 1) 22k k=1

(8)

Miscellaneous Results

557

γ = 2 − 2 log 2 −

∞ X ζ (2k + 1) − 1 . (k + 1)(2k + 1)

(9)

k=1

∞

γ = 1+

8 X ζ (2k + 1) − 1 1 log − 3 15 2k + 1 k=1

 2k 3 . 2

(10)

∞

X ζ (k) − 1 k−1 1 γ = 1 − log 6 + (−1)k 2 . 2 k

(11)

k=2

γ = 1−

∞ X

ζ (2k + 1) , (k + 1)(2k + 1)

k=1

(12)

which is commented on in the work of Ramanujan [964], who gave a general formula containing (11) as a very special case. γ = log 2 − 2

∞ X

Ak X

k

k=1 j=Ak−1 +1

1 (3j)3 − 3j

,

(13)

where 1 Ak := (3k − 1) 2

(k = 0, 1, 2, . . .). k

∞ 2 −1 1 1 X X (−1)i γ= − + k . 2 3 2k + i

(14)

i=0

k=2

k

∞ 2 −1 1 X X γ= + k 12i(2i + 1)(2i + 2). 2 k−1 k=1

γ = 1−

∞ X k=1

(15)

i=2

k

k

2 X i=2k−1 +1

1 . (2i − 1)(2i)

(16)

Campbell [208, p. 200] recorded an interesting series representation for γ that can easily be written in the form: ∞

γ = 1 − log

3 X ζ (2k + 1) − 1 − . 2 (2k + 1)2k

(17)

k=1

We note that the expression of γ in (17) seems to be the most rapidly convergent series among the ever-known series representations of γ , by observing 1 < ζ (2k + 1) < 2 for each positive integer k and the following rough estimations: 0<

∞ X ζ (2k + 1) − 1 < 10−65 (2k + 1)2k

k=21

and

0<

∞ X ζ (2k + 1) − 1 < 10−153 . (2k + 1)2k

k=41

558

Zeta and q-Zeta Functions and Associated Series and Integrals

However, when (17) is compared with (8), it is carefully concluded that the expression (17) is incorrectly recorded. Choi and Nash [277] obtained a class of integral representations of the GlaisherKinkelin constant A, defined by 1.4(2): 1 1 log A = + 8 24

−



2 3+ 2 p −1 

 log p h i j p



 Z  (p−1)/2  X  1 j   t − {ψ(1 + t) + ψ(1 − t)} dt j log j + p ,    p p2 − 1 j=1

(18)

0

where p is an odd positive integer greater than 1. By using the theory of the double Gamma function 02 , the following log-Gamma integral was evaluated by Choi and Srivastava [289, p. 100, Eq. (2.29)] (see also 3.4(496)): 1

Z2

1 7 1 3 log 0(t + 1) dt = − − log 2 + log π + log A, 2 24 4 2

(19)

0

which was equivalent to 2.2(37). Gosper [498, p. 71] made use of the Polylogarithm function (cf. [752]) to evaluate the log-Gamma integral (19) in the form: 1

Z2 0

1 γ 1 3 3 ζ 0 (2) log 0(t + 1) dt = − + − log 2 + log π − , 2 8 6 8 4π2

(20)

so that, by comparing (19) and (20), we obtain the following relationship among π, γ , A and ζ 0 (2):   1 0 2 γ ζ (2) = π + log(2π) − 2 log A , (21) 6 6 or, equivalently, γ=

6 ζ 0 (2) A12 + log . 2π π2

(22)

Conversely, if we apply the known relationship 2.1(31) in (21), we find that (see [294, p. 240, Eq. (18)]) ζ 0 (−1) =

1 ζ 0 (2) [1 − γ − log(2π)] + , 12 2π2

(23)

Miscellaneous Results

559

which can also be obtained by appealing appropriately to Riemann’s functional equation for ζ (z) (see 2.3(11)). The relationship (22) is recorded erroneously in the work of Voros [1201, p. 453, Eq. (6.25)]. Bendersky [114] (see also [6, p. 199]) presented a set of constants, including B and C, defined, respectively, by 1.3(69) and 1.3(70): there exists constants Dk , defined by

log Dk := lim

n→∞

n X

! (k ∈ N0 ) ,

k

m log m − p(n, k)

(24)

m=1

where the definition of p(n, k) in Adamchik [6, p. 198, Eq. (20)] is corrected here as follows:   nk+1 1 nk log n + log n − p(n, k) : = 2 k+1 k+1   j k k−j X X
n Bj+1 1 log n + 1 − δkj  + k! ( j + 1)! (k − j)! k−`+1 `=1

j=1

and δkj is the Kronecker symbol, defined by δkj = 0 (k 6= j) and δkj = 1 (k = j). For the constants Dk (k ∈ N0 ), defined in (24), we can show that 1

D0 = (2π) 2 ,

D1 = A,

D2 = B and D3 = C

and log Dk =

Bk+1 Hk − ζ 0 (−k) k+1

(k ∈ N0 ) ,

(25)

where Bn are the Bernoulli numbers and Hn are the harmonic numbers. The constants introduced in this section can be seen to be involved in the theory of multiple Gamma functions. For example, the following log-multiple Gamma integral (see [269, p. 523, Eq. (2.50)]) 3

Z2

log 03 (t + 2) dt =

0

Z1

Z1 log G(t + 1) dt +

0

Z + 0

1 2

log 0(t + 1) dt + 2

log 03 (t + 1) dt

0

Z

1 2

1

Z2 log G(t + 1) dt +

0

(26) log 03 (t + 1) dt

0

259 29 9 15 5 15 =− − log 2 + log π − log A − log B + log C. 768 1920 16 16 4 16

560

Zeta and q-Zeta Functions and Associated Series and Integrals

A Class of Constants Analogous to {Dk } Setting f (x) = 1x , a = 1, b = N, K = 1 in the Euler-Maclaurin summation formula 2.7(21) and taking the limit of the resulting equation as N → ∞, we obtain the EulerMascheroni constant γ given in 1.1(3). By appealing to the Leibniz’s rule for differentiation, we find that the mth order derivative of f (x) := xk−1 log x (k ∈ N) is f (m) (x) = αk,m xk−m−1 log x + βk,m xk−m−1


0 5 m 5 k − 1; m ∈ N0 ,

(27)

where the constants αk,m and βk,m are given as follows: αk,m =

(k − 1)! (k − m − 1)!

and βk,m =

m (k − 1)! X 1 (k − m − 1)! k−m−1+j j=1
0 5 m 5 k − 1; m ∈ N0 .

(28)

Note also that f (k) (x) =

αk,k−1 x

and f (k+1) (x) = −

αk,k−1 . x2

(29)

We prove the following identity: k−1 X (−1)m m=1

m!

Bm βk,m−1 =

k   (−1)k X k 1 1 Bk−m + 2 k m m k

(k ∈ N),

(30)

m=1

the empty sum being interpreted (as usual, in what follows) to be nil. Indeed, since (30) holds trivially for k = 1, we assume k ∈ N \ {1}. Let Sk := (−1)k k

k−1 X (−1)m m=1

m!

Bm βk,m−1

(k ∈ N \ {1}),

which, upon setting n = k − m and using (28), yields Sk =

k−1 X

(−1)n

n=1

  k−n−1 X 1 k Bk−n . n n+j

(31)

j=1

Define a function H(x) given by H(x) :=

k−1 X n=1

(−1)n

  k−n−1 X k Bk−n xn−1+j . n j=1

(32)

Miscellaneous Results

561

Since the innermost sum is a finite geometric series and Z1 Sk =

H(x) dx,

0

we find that H(x) =

h(x) − xk−1 h(1) , 1−x

(33)

where, for convenience,

h(x) :=

k−1   X k n=1

n

Bk−n (−x)n .

It follows from Equations 1.6(3) and 1.6(4) that h(x) = Bk (−x) − Bk − (−x)k = Bk (1 − x) − Bk − k(−x)k−1 − (−x)k ,

(34)

which, upon setting x = 1, yields h(1) = (−1)k (k − 1).

(35)

Putting (34) and (35) into (33), we get H(x) =

Bk (1 − x) − Bk + (−1)k xk−1 , 1−x

which, upon integrating from 0 to 1 and changing the variable t = 1 − x, gives Z1 Sk =

(−1)k Bk (t) − Bk dt + . t k

(36)

0

Applying Equation 1.7(3) to (36), we obtain

Sk =

k   X k 1 1 Bk−m + (−1)k , k m m

m=1

which, upon dividing by (−1)k k, leads to the desired identity (30).

(37)

562

Zeta and q-Zeta Functions and Associated Series and Integrals

Setting f (x) = xk−1 log x (k ∈ N), a = 1, b = N (N ∈ N \ {1}) and K = k + 1 in 2.7(21) and using (30), we can obtain a class of mathematical constants {Ck }k∈N , defined for k ∈ N by " Ck := lim

N→∞

−

k X m=1

N X

nk−1 log n −

n=1

Nk Nk log N + 2 k k

# (38) k−1 X (−1)m (−1)m k−m k−m Bm αk,m−1 N log N − Bm βk,m−1 N , m! m! m=1

where αk,m and βk,m are defined by (28) (see [274]). Remark 1 In the process of getting the constants Ck in (38), we find that k   (−1)k (−1)k X k 1 Bk−m + Bk+1 Ck = − k m m k(k + 1) m=1 Z∞

−

(−1)k

k(k + 1)

Bk+1 ({x}) dx x2

(39) (k ∈ N).

1

√ Remark 2 It follows from the Stirling’s formula 1.1(32) for n! that C1 = log 2π . Note also that C2 = log A, where A is the Glaisher-Kinkelin constant given in 1.4(2), C3 = log B and C4 = log C, where the constants B and C are given in 1.4(69) and 1.4(70), respectively. Remark 3 We find that the constants {Ck }k∈N are connected with BenderskyAdamchik constants {Dk }k∈N in (24) as follows: log Dk−1 = Ck (k ∈ N).

(40)

Remark 4 As a matter of fact, Choi and Lee [274] introduced the constants {Ck }k∈N in (38) to evaluate the following family of series associated with the Riemann zeta function:   k ∞ X X 1 γ k (−1)m ` ζ (m) = + + (−1) C` m+k k k+1 `−1

m=2

(k ∈ N) ,

(41)

`=1

which may be evaluated as a special case of a generalized theorem very recently developed [297, Theorem 7.2, p. 403].

Miscellaneous Results

563

Other Classes of Mathematical Constants Choi et al. [310, p. 112, Eq. (36)] introduced a class of mathematical constants Ap (p > 0), defined by " n     X 1 1 log k + log Ap := lim k+ n→∞ p p k=1 (42)  2       1 1 1 1 n2 n n 1 1 − + + n+ 2 + + log n + + + , 2 2 p 2p 12 p 4 2p 2p by making use of the Euler-Maclaurin summation formula 1.4(68). It is easy to see that (cf. [310, p. 114]) 1 log A1 = log A − , 4

(43)

where A denotes the Glaisher-Kinkelin constant. Choi et al. [310, p. 112, Eq. (37)] also presented another class of mathematical constants Cp (p > 1): " n     X 1 1 log Cp := lim k− log k − n→∞ p p k=1       2  1 1 1 1 1 n2 n 1 n + − n+ 2 − + log n − + − , − 2 2 p 2p 12 p 4 2p 2p

(44)

where (obviously) Cp = A−p .

(45)

In terms of the mathematical constants Ap and Cp , the following log-Gamma integrals were evaluated earlier by Choi et al. [310, p. 114, Eqs. (39) and (40)]: Z1/p 1 1 3 log 0(1 + t) dt = log A − log Ap + log(2π) − − 2 2p 2p 4p

(p > 0)

(46)

(p > 1).

(47)

0

and Z1/p 1 1 3 log 0(1 − t) dt = log Cp − log A + log(2π) − + 2 2p 2p 4p 0

With a view to deriving a relationship between   1 0 Ap and ζ −1, (p > 0) p

564

Zeta and q-Zeta Functions and Associated Series and Integrals

analogous to that in 2.1(31), we set 

1 f (x) := x + p

−s

(p > 0)

in the Euler-Maclaurin summation formula 1.4(68) with a = 0. We, then, obtain n  X k=1

1 k+ p

−s ∼ C(s, p) +

 1−s n + 1p − ps−1

+

1 2

 n+

1 p

−s

1−s   m X B2r 1 −s−2r+1 − (s)2r−1 n + + R(s, p; n; m) (2r)! p

(48)

r=1

(<(s) > −2m − 1; m ∈ N0 ), where C(s, p) is a constant depending on s and p, (s)n denotes the Pochhammer symbol defined by 1.1(5) and the remainder part R(s, p; n; m) satisfies the following limit relationship: lim R(s, p; n; m) = 0

n→∞

(<(s) > −2m − 1; m ∈ N0 ) .

In fact, C(s, p) can be expressed explicitly as follows, by applying the same f (x) to 1.4(68) and reducing the domain to <(s) > 1:    1−s 1 s−1   n −s n + − p p 1 X  C(s, p) = lim  k+ −  n→∞ p 1−s k=1 (49)   s−1 1 p = ζ s, − ps − (<(s) > 1). p s−1 Now, it follows from (48) and (49) that   1−s   −s   n  n + 1p X 1 1 1 1 −s  ζ s, = lim  k+ − − n+ n→∞ p p 1−s 2 p k=1

  −s−2r+1 m X 1 B2r  s + (s)2r−1 n + +p (2r)! p r=1

(m ∈ N, <(s) > −2m − 1, and s 6= 1; m = 0 and <(s) > 1). We note that the right-hand side of (50) is analytic for <(s) > −2m − 1

(m ∈ N0 )

and s 6= 1.

(50)

Miscellaneous Results

565

Thus, we can differentiate (50) with respect to s under the limit sign and set s = −1 in the resulting equation. We, therefore, obtain (see [296, Eq. (3.24)])  1−s " n  1  −s    n + X p 1 1 1 k+ = lim − log k + − ζ 0 s, n→∞ p p p (s − 1)2 k=1  1−s       n + 1p 1 1 1 −s 1 + log n + + n+ log n + 1−s p 2 p p       2r−1 m X B2r 1 −s−2r+1  X 1 1  (s)2r−1 n + − log n + +  (2r)! p s+j−1 p  r=1 s

+ p log p

j=1

(m ∈ N, <(s) > −2m − 1, and s 6= 1; m = 0 and <(s) > 1), (51)

where we have also used the following elementary identity: ! n X d 1 (s)n = (s)n (n ∈ N0 ) . ds s+k−1

(52)

k=1

By setting s = −1 and s = −2 in (51), we find that " n       X 1 1 1 0 = − lim k+ log k + ζ −1, n→∞ p p p k=1  2      1 n 1 1 1 1 1 (53) − + + n+ 2 + + log n + 2 p 2 2p 12 p 2p # n 1 1 1 n2 + − 2+ + log p (p > 0) + 4 2p 12 p 4p and " n        3   X 1 1 1 2 1 n 1 2 ζ −2, = − lim log k + − + + n k+ n→∞ p p p 3 p 2 k=1      1 1 1 1 1 1 1 n3 + 2+ + n+ 3 + 2 + log n + + p 6 6p p 9 p 3p 2p    2 n 1 1 1 1 log p + + − n − 3+ + 2 (p > 0). (54) 2 3p 12 12p 3p 9p p 0

Comparing (42) and (53), we finally obtain the desired relationship:   1 1 1 1 0 log Ap = −ζ −1, + − 2 + log p (p > 0), p 12 4p p which, in view of (43), immediately yields 2.1(31) when p = 1.

(55)

566

Zeta and q-Zeta Functions and Associated Series and Integrals

Next, in view of the well-known reflection formula (see 1.1(12)): 0(1 + z) 0(1 − z) =

πz , sin(π z)

we can evaluate the following integral (cf. [310, p. 114, Eq. (43)]):   Zπ/p 1 log sin t dt = π log Ap − log Cp − log(2p) p

(p > 1),

(56)

0

which, by virtue of another known result [289, p. 95, Eq. (2.2)]:       π 1 Zπ/p sin G 1 + p p π  + π log    log sin t dt = log  1 p 2π G 1 − p 0

(57)

yields the following relationship between the mathematical constants Ap and Cp :    1/p G 1 + 1 p π p   Cp sin Ap = 1 π p G 1− p 

(p > 1).

(58)

To get a more general class of mathematical constants (see [296]) than those given by (42), we begin by differentiating the function     1 1 q log x + f (x) := x + p p

(p > 0; q ∈ N)

(59)

l times (l ∈ N). We, thus, obtain    q−l  Y   l  1  (q − j + 1) log x + 1 + Pl (q) f (l) (x) = x +   p p 

(l ∈ N),

j=1

(60) where Pl (q) is a polynomial of degree l − 1 in q satisfying the following recurrence relation:

Pl (q) :=

 l−1 Y    (q − l + 1) Pl−1 (q) + (q − j + 1)

(l ∈ N \ {1}),

j=1

  

1

(l = 1).

(61)

Miscellaneous Results

567

In fact, by mathematical induction on l ∈ N, we can give an explicit expression for Pl (q) as follows:   l l X Y Pl (q) =  (q − j + 1)  j=1

j=1

 1  q−j+1

(l ∈ N).

(62)

By substituting from (59) and (60) into the Euler-Maclaurin summation formula 1.4(68) with a = 0, we get a class of mathematical constants Cp,q , defined by "

n  X

     1 q 1 1 1 q+1 log Cp,q := lim k+ log k + + n+ n→∞ p p p (q + 1)2 k=1  q+1    q   1 1 1 1 1 1 − n+ log n + − n+ log n + q+1 p p 2 p p   [(q+1)/2] X B2r 1 q−2r+1 n+ − (2r)! p r=1       2r−1  Y 1 ·  (q − j + 1) log n + + P2r−1 (q)  (p > 0; q ∈ N),   p

(63)

j=1

where [x] denotes (as usual) the greatest integer less than or equal to x. Setting q = 1 in (63) and comparing the resulting equation with (42), it is easy to get a relationship between Ap and Cp,1 as follows: 

1 1 Cp,1 = exp − 2 12 4p

 Ap

(p > 0).

(64)

If we set s = −q (q ∈ N) in (51), we obtain " n         X 1 1 q 1 1 1 q+1 −ζ −q, = lim log k + + n + k+ n→∞ p p p p (q + 1)2 k=1         1 q+1 1 1 1 q 1 1 n+ log n + − n+ log n + − q+1 p p 2 p p       m 2r−1 X B2r 1 q−2r+1  Y 1 − n+ (q − j + 1) log n +  (2r)! p p r=1 j=1    2r−1 2r−1  Y X 1   − p−q log p + (q − j + 1)  q−j+1  0

j=1

j=1

(m, q ∈ N; q < 2m + 1).

(65)

568

Zeta and q-Zeta Functions and Associated Series and Integrals

Thus, by comparing (63)  and (65) and applying (62), we get the following relationship 0 between Cp,q and ζ −q, p1 :   1 log Cp,q = −ζ −q, + p−q log p (p > 0; q ∈ N). p 0

(66)

We conclude this section by remarking that, in a mild sense, the constants Cp,q are generalizations of the Bendersky-Adamchik constants Dk in (24), because, in view of (25) and (66), there is a relationship between Dk and C1,k : log C1,k = log Dk −

Bk+1 Hk k+1

(k ∈ N).

(67)

7.2 Log-Sine Integrals Involving Series Associated with the Zeta Function and Polylogarithms Motivated essentially by their potential for applications in a wide range of mathematic and physical problems, the Log-Sine integrals have been evaluated, in the existing literature on the subject, in many different ways. Very recently, Choi et al. [270] showed how nicely some general formulas analogous to the generalized Log-Sine integral
(m) Lsn π3 can be obtained using the theory of Polylogarithms. Relevant connections of the results presented here with those obtained in earlier works are also indicated precisely. For the Log-Sine integrals Lsn (θ) of order n, defined by (see 2.4(82)) Zθ  x n−1 Lsn (θ) := − log 2 sin dx 2

(n ∈ N \ {1}),

(1)

0

the recurrence relation 2.4(111) holds true when θ = π (see, e.g., [752, p. 218, Eq. (7.112)]). By using an idea analogous to that of Shen [1024], Beumer [128] presented a recursion formula for (−1)n−1 D(n) := 2 · (n − 1)!

Zπ h  x in−1 dx log sin 2

(n ∈ N)

0

in the following form: 2n−1 X k=1

(−1)k−1 D(k) D(2n − k) = (−1)n+1

22n − 1 2n π B2n (2n)!

(n ∈ N),

(2)

Miscellaneous Results

569

where Bn are the Bernoulli numbers (see Section 1.7), and D(1) =

π 2

and D(2) =

π log 2. 2

In fact, by mainly analyzing the generalized binomial theorem and the familiar Weierstrass canonical product form of the Gamma function 0(z) (see 1.1(2)), Shen [1024, p. 1396, Eq. (19)] evaluated the Log-Sine integral Lsk+1 (2π) as follows: 1 2π

Z2π  0

∞ x k k! X n log 2 sin dx = (−1)k k σk 2 2 k

(k, n ∈ N),

(3)

n= 2

where σkn are given in terms of the Stirling numbers s(n, k) of the first kind (see, for details, Section 1.5; see also [969]), by σkn =

k−1 X s(n, k − m) s(n, m) . n! n!

m=1

More recently, Batir [102] presented integral representations, involving Log-Sine terms, for some series associated with  −1  −2 2k 2k −n k and k−n , k k and for some closely-related series, by using a number of elementary properties of Polylogarithms. Lewin [752, pp. 102–103; p. 164] presented the following integral formulas: π

Z2

 x dx = −G log 2 sin 2

(4)

 35 1 x dx = ζ (3) − π G, x log 2 sin 2 32 2

(5)

0

and π

Z2 0

where G denotes the Catalan constant, defined by 1.4(16). Several other authors have concentrated on the problem of evaluation of the Log(m) Sine integral Lsn (θ) and the generalized Log-Sine integral Lsn (θ) of order n and index m, defined by Ls(m) n (θ) := −

Zθ 0

 x n−m−1 xm log 2 sin dx 2

(6)

570

Zeta and q-Zeta Functions and Associated Series and Integrals

with the argument θ given by θ = π3 . (Throughout this section, we choose the principal branch of the logarithm function log z in case z is a complex variable.) In particular, van der Poorten [1181] proved that π

Z3

 x 7 3 log2 2 sin π dx = 2 108

(7)

 x 17 4 x log2 2 sin dx = π . 2 6480

(8)

0

and π

Z3 0

Zucker [1267] established the following two integral formulas: π

 Z3    x 3 2 x 253 5 4 2 log 2 sin − x log 2 sin dx = π 2 2 2 3240

(9)

0

and π

 Z3    x  x3 x 313 x log4 2 sin − log2 2 sin dx = π 6. 2 2 2 408240

(10)

0



(m) Zhang and Williams [1253] extensively investigated Lsn π3 and Lsn π3 along with other integrals to present two general formulas (see [1253, p. 272, Eqs. (1.6) and (1.7)]), which include the integral formulas (7) to (10) as special cases. We choose to recall here one more explicit special case of the Zhang-Williams integral formulas as follows: π

Z3    x  15 2 x log6 2 sin − x log4 2 sin 2 4 2 0   15 x 77821 + x4 log2 2 sin dx = 6 6 π 7 . 16 2 2 ·3 ·7

(11)

The following well-known formula is recorded (see [531, p. 334, Entry (50.5.16)]):

x · cot x = 1 +

∞ X n=1

(−1)n

22n B2n 2n x (2n)!

(|x| < π).

(12)

Miscellaneous Results

571

Analogous Log-Sine Integrals Choi et al. [270] showed how nicely some general formulas analogous to the gener
(m) alized Log-Sine integral Lsn π3 can be obtained, by using the theory of Polylogarithms. Indeed, by carrying out repeated integration by parts in 2.4(71) in conjunction with 2.4(3), we obtain

Lin (z) − Lin (w) =

Zz

Lin−1 (t)

dt t

w n−2 X (−1)n−1 (−1)k−1 = (log t)k Lin−k (t) + (log t)n−1 log(1 − t) k! (n − 1)! k=1

+

(−1)n−1 (n − 1)!

Zz

(log t)n−1

w

dt 1−t

(n ∈ N \ {1}),

! z

t=w

(13)

where (and elsewhere in this section) an empty sum is understood to be nil; in particular, we have (−1)n−1 ζ (n) = Lin (1) = (n − 1)!

Z1

(−1)n−1

Z1

=

(n − 1)!

(log t)n−1

0

dt 1−t (14)

[log(1 − t)]n−1

dt t

(n ∈ N \ {1}),

0

which, by substituting t = u−1 , yields

(−1)

n−1

Ze−iθ

(log t)

1

n−1

dt = 1−t

Zeiθ 1

(log t)n−1

dt 1 + (iθ)n 1−t n

(15)

(0 5 θ 5 π). Furthermore, it is easily observed, by setting t = 1 − eix , that 1−e Z iθ 0

(log t)

n−1

dt = −i 1−t

Zθ  0

 1 x n−1 i (x − π) + log 2 sin dx, 2 2

(16)

572

Zeta and q-Zeta Functions and Associated Series and Integrals

or, equivalently, that 1−e Z iθ

(log t)

n−1

1

dt = −i 1−t

Zθ 

 1 x n−1 i (x − π) + log 2 sin dx 2 2

(17)

0

+ (−1)n (n − 1)! ζ (n)

(n ∈ N \ {1})

in view of (14). In its special case when n = 2m + 1 (m ∈ N) and w = 1, (13) yields Zz

(log t)2m

1

dt = (2m)! Li2m+1 (z) − (2m)! ζ (2m + 1) 1−t + (2m)!

2m−1 X k=1 2m

− (log z)

(18)

(−1)k (log z)k Li2m+1−k (z) k!

log (1 − z)

(m ∈ N).

π

Putting z = ei 3 in (18) and using the following elementary identity: π

π

1 − ei 3 = e−i 3 ,

(19)

we get i π3

Ze

(log t)2m

1

 π 2m+1 dt = i (−1)m − (2m)! ζ (2m + 1) 1−t 3 + (2m)!

2m−1 X k=0

(−1)k k!

 π k  π i Li2m+1−k ei 3 3

(20) (m ∈ N).

We, now, separate the even and odd parts of the sum occurring in (20) and make use of 2.4(79) and 2.4(85). We, thus, obtain 2m−1 X k=0

 π  2m−1 π  X (−1)[k/2]  π k (−1)k  π k i Li2m+1−k ei 3 = Cl2m+1−k k! 3 k! 3 3

+i

k=0

2m−1 X k=0

h

(−1) k!

k+1 2

i

 π k 3

Gl2m+1−k

π  3

.

(21)

Upon substituting from (21) into (20), and equating the real and imaginary parts on each side of the resulting equation, we obtain

Miscellaneous Results

573

i π3



Ze

 < 

 (log t)2m

1

dt   = −(2m)! ζ (2m + 1) 1−t (22)

+ (2m)!

2m−1 X k=0

π  (−1)[k/2]  π k Cl2m+1−k k! 3 3

(m ∈ N)

and   = 

i π3

Ze

 (log t)2m

1

 2m+1 dt   = (−1)m π 1−t 3 (23)

+ (2m)!

2m−1 X k=0

h

(−1) k!

k+1 2

i

 π k 3

Gl2m+1−k π 3

Setting n = 2m + 1 (m ∈ N) and θ = −i π3

eZ

π  3

(m ∈ N).

in (15), we have

i π3

(log t)2m

1

dt = 1−t

Ze

(log t)2m

1

+

(−1)m 2m + 1

dt 1−t

(24)

 π 2m+1 3

Also, by putting n = 2m + 1 (m ∈ N) and θ =

π 3

i

(m ∈ N).

in (17), and using (19), we obtain

−i π3

eZ

(log t)2m

1

dt = −(2m)! ζ (2m + 1) 1−t (25)

π

Z3  −i

  x  2m 1 i (x − π) + log 2 sin dx 2 2

(m ∈ N).

0

Now, by using the binomial theorem, we find that    x  2m 1 i (x − π) + log 2 sin 2 2     m  X 2m x − π 2k x = (−1)k log2m−2k 2 sin 2k 2 2 k=0     m  X 2m x − π 2k−1 x +i (−1)k+1 log2m+1−2k 2 sin 2k − 1 2 2 k=1

(26) (m ∈ N).

574

Zeta and q-Zeta Functions and Associated Series and Integrals

Substituting (26) into the integrand on the right-hand side of (25) and equating the real and imaginary parts of each side of the resulting equation, we obtain   < 

−i π3



eZ

(log t)2m

dt   = −(2m)! ζ (2m + 1) 1−t

(−1)k+1



1

(27)

π

+

m X

22k−1

k=1

 Z3

2m 2k − 1

 x dx (x − π )2k−1 log2m+1−2k 2 sin 2

(m ∈ N)

0

and   = 

−i π3

eZ

 dt   1−t

(log t)2m

1

(28)

π

  Z3 m  X (−1)k+1 2m x 2k 2m−2k = dx − π log 2 sin (x ) 2k 2 22k k=0

(m ∈ N).

0

Finally, from (22), (23), (24), (27) and (28), we can deduce the following analogous Log-Sine integral formulas:   Zπ/3 m  X (−1)k+1 x 2m dx (x − π )2k−1 log2m+1−2k 2 sin 2k−1 2k − 1 2 2 k=1

0

= (2m)!

2m−1 X k=0

(29)

π  (−1)[k/2]  π k Cl2m+1−k k! 3 3

(m ∈ N)

and π

  Z3 m  X (−1)k+1 2m x 2k 2m−2k − π) log 2 sin dx (x 2k 2 22k k=0

0

 m = (−1) 1 +

+ (2m)!

1 2m + 1

2m−1 X k=0

h

(−1) k!

  π 2m+1 3 k+1 2

(30)

i

 π k 3

Gl2m+1−k

π  3

(m ∈ N).

Miscellaneous Results

575

Just as in our derivations of (29) and (30), if we set π

n = 2m (m ∈ N), z = ei 3

and w = 1

in (13) and apply (15) and (17) with θ = π3 , we obtain some other analogous Log-Sine integral formulas as follows: π

  Z3  2m − 1 x 2k+1 2m−2−2k dx − π log 2 sin (x ) 2 22k+1 2k + 1 k=0 0   1  π 2m = (−1)m+1 1 + − 2 (2m − 1)! ζ (2m) 2m 3

m−1 X

(−1)k

+ (2m − 1)!

2m−2 X k=0

(31)

h i

(−1) k!

k 2

 π k 3

Gl2m−k

π  3

(m ∈ N)

and π

m−1 X k=0

  Z3  (−1)k 2m − 1 x 2k 2m−1−2k dx − π log 2 sin (x ) 2 2k 22k

= (2m − 1)!

0 2m−2 X k=0

(32) h

(−1) k!

k−1 2

i

 π k 3

Cl2m−k

π  3

(m ∈ N).

Remarks on Cln (θ ) and Gln (θ ) In view of Equations (29) to (32), we need to express the generalized Clausen function Cln (θ) and its associated Clausen function Gln (θ) as explicitly as possible, at least at the argument θ = π/3. To do this, we begin by rewriting a known formula ∞ ∞ X X sin(2πkx) cos(2πkx) + i n k kn k=1 k=1   (2πi)n−1 0 Bn (x) n−1 0 = ζ (1 − n, x) + (−1) ζ (1 − n, 1 − x) − πi (n − 1)! n (0 < x < 1; n ∈ N \ {1}),

(33)

which was proven by Adamchik [6, Eq. (9)] (see also [829, Eq. (21)]), who used Lerch’s transform formula [745] and where Bn (x) are the Bernoulli polynomials of degree n in x (see, for details, Section 1.7). Now, replacing n by 2n and 2n + 1 in (33) and equating the real and imaginary parts in each case, we obtain the following

576

Zeta and q-Zeta Functions and Associated Series and Integrals

formulas 2.4(109) and 1+

Cln (2πx) = (−1)

h

1 2 (n+1)

i

o (2π)n−1 n 0 ζ (1 − n, x) + (−1)n+1 ζ 0 (1 − n, 1 − x) (n − 1)! (0 < x < 1; n ∈ N \ {1}). (34)

Srivastava et al. [1097, Eq. (3.8) and Eq. (3.17)] presented the following formulas:

Cl2n+1

π  3

=

  1 1 − 2−2n 1 − 3−2n ζ (2n + 1) 2

(n ∈ N)

(35)

and Cl2n

√         3 1 1 =, 2n ζ 2n, + ζ 2n, − 22n−1 32n − 1 ζ (2n) 3 3 6 6

π 

(n ∈ N). (36)

Cvijovic´ and Klinowski [349, Eq. (10a)] proved formulas that can be specialized in the following form:   q 1 X k ζ n, qn q k=1      1 − (−1)n 1 + (−1)n 2kπp 2kπp sin + cos 2 q 2 q (n, q + 1 ∈ N \ {1}; p ∈ Z), 

Cln

2π p q



=

(37)

which, upon replacing n by 2n + 1 and 2n with p = 1 and q = 6, leads to (35) and (36), respectively. We are ready to consider some explicit expressions of (29) to (32). Upon setting m = 1, 2, 3 in (30) and m = 2, 3 in (31), if we apply 2.4(109), we obtain the following explicit analogous Log-Sine integral formulas: π

Z3

 7 3 x dx = π , log2 2 sin 2 108

(38)

 x 403 dx = − 4 4 π 4 , (x − π) log2 2 sin 2 2 ·3 ·5

(39)

0 π

Z3 0 π

Z3  0

   3 x x 73 2 2 4 − log 2 sin dx = 4 4 π 5 , (x − π) log 2 sin 2 2 2 2 ·3 ·5

(40)

Miscellaneous Results

577

π

Z3 h   x 39883 x i 2 (x − π) log4 2 sin π 6, − (x − π )3 log2 2 sin dx = − 4 6 2 2 2 ·3 ·5·7 0

(41) and π

Z3    x  15 x log6 2 sin − (x − π )2 log4 2 sin 2 4 2 0   15 x 697 + dx = 6 6 π 7 . (x − π )4 log2 2 sin 16 2 2 ·3 ·7

(42)

It should be noted that the integral formulas (7) to (11) can easily be deduced from these last integral formulas (38) to (42). Upon setting m = 1, 2 in (29) and (30), if we apply (35) and (36), we obtain π

Z3

√ √       x 2 4 3 3 1 1 3π (x − π) log 2 sin dx = ζ (3) − π + ζ 2, + ζ 2, . 2 3 81 54 3 6

0

(43) π 3

 Z    x 1 x 3 3 2(x − π) log 2 sin − (x − π) log 2 sin dx 2 2 2 0 √ √      100 4π 2 8 3 5 3π 1 1 = ζ (5) − ζ (3) − π + ζ 4, + ζ 4, 9 9 243 162 3 6 √ 3     3π 1 1 − ζ 2, + ζ 2, . 243 3 6 π

Z3

√ √       x 2 3 2 3 1 1 dx = π − log 2 sin ζ 2, + ζ 2, . 2 27 36 3 6

(44)

(45)

0 π

 Z3    x 3 x log3 2 sin − (x − π)2 log 2 sin dx 2 4 2 0 √ √ 2     2 3 4 2π 3π 1 1 = π + ζ (3) + ζ 2, + ζ 2, 243 3 108 3 6 √      3 1 1 − ζ 4, + ζ 4, . 216 3 6

(46)

578

Zeta and q-Zeta Functions and Associated Series and Integrals

Further Remarks and Observations We consider the following general integral: (m) Lsm+2 (z) =

Zz

 x xm log 2 sin dx 2

(m ∈ N0 ).

(47)

0

Applying integration by parts in (47) and using (12) and 2.3(18), we obtain Zz 0

  zm+1 z zm+1 x dx = log 2 sin − xm log 2 sin 2 m+1 2 (m + 1)2 ∞ 2 zm+1 X ζ (2k)  z 2k + m+1 m + 2k + 1 2π

(48) (|z| < 2π; m ∈ N).

k=1

If the known formulas Eq. (2.16) in [269, p. 515] and Eq. (2.13) in [269, p. 514] are used, the infinite series in (48) can be expressed as finite series as follows: Zz

 2m+1    x z2m+1 z  (2π)2m+1 X 2m + 1 x2m log 2 sin dx = log 2 sin + 2 2m + 1 2 2m + 1 k k=0

0

 h  z i  z 2m+1−k z  + (−1)k ζ 0 −k, 1 + · ζ 0 −k, 1 − 2π 2π 2π (|z| < 2π; m ∈ N0 )

(49)

and Zz

  x z2m z x2m−1 log 2 sin dx = log 2 sin + (−1)m+1 (2m − 1)! ζ (2m + 1) 2 2m 2

0

+

 2m   (2π)2m X 2m h 0  z  z i  z 2m−k ζ −k, 1 − + (−1)k ζ 0 −k, 1 + k 2m 2π 2π 2π k=0

(|z| < 2π; m ∈ N).

(50)

The special cases of (49) when m = 0 and (50) when m = 1 would, in light of some of the identities in Chapter 2, readily yield, respectively, the following formulas: π

Z3 0

     π  x 5 1 0 0 log 2 sin dx = 2π ζ −1, − ζ −1, = −Cl2 2 6 6 3 !
02 65 π = − log(2π) + 2π log
3 02 67 

(51)

Miscellaneous Results

579

and π

Z3 0

     2ζ (3) 2π 2 0 5 1 x 0 dx = + ζ −1, − ζ −1, x log 2 sin 2 3 3 6 6
! 02 65 2ζ (3) π 2 2π 2 = − log(2π) + log
, 3 9 3 02 67 

(52)

where 02 denotes the double Gamma function (see [94, 96]; see also Section 1.4). Here, it should be remarked in passing that, in view of Equation (34) and Equations (49) to (52), closed-form expressions for derivatives of ζ (s, a) at the negative integer values of s and rational values of a are needed, some of which were given by Adamchik [6] and by Miller and Adamchik [829]. We note also that Srivastava et al. [1097] studied extensively and systematically some definite integrals in conjunction with series involving the Zeta function, such as in (48). If (34) is used, (49) and (50) are readily rewritten as follows: Zz

2m+2  X (−1)k+[ 21 (k+1)] x x2m log 2 sin dx = (2m)! Clk (z) z2m+2−k 2 (2m + 2 − k)! k=2

0

(53)

(|z| < 2π; m ∈ N0 ) and Zz

 x x2m−1 log 2 sin dx = (−1)m+1 (2m − 1)! ζ (2m + 1) 2

0

+ (2m − 1)!

2m+1 X k=2

(54)

(−1)k+[ 2 (k+1)] Clk (z) z2m+1−k (2m + 1 − k)! 1

(|z| < 2π; m ∈ N). By setting m = 1 in (53) and m = 2 in (54) with z = π3 , we obtain π

Z3 0

√  x 2 3 4 2π x log 2 sin dx = − π − ζ (3) 2 729 9 √ 2     √      3π 1 1 3 1 1 − ζ 2, + ζ 2, + ζ 4, + ζ 4, 324 3 6 648 3 6 2

(55)

580

Zeta and q-Zeta Functions and Associated Series and Integrals

and π

√  x 2 3 5 π2 29 x log 2 sin dx = − π − ζ (3) − ζ (5) 2 243 9 9 √ 3     √      1 1 1 1 3π 3π ζ 2, + ζ 2, + ζ 4, + ζ 4, . − 972 3 6 648 3 6

Z3

3

0

(56)

By using 1.3(53), Shen’s process (see [1024, pp. 1393–1394]) can be shortened considerably and it may also be easier to apply to other situations of a similar nature (see Section 3.5). For example, the formula (3) can be stated as follows: 1 2π

Z2π h  x in (−1)n n! log 2 sin dx = an 2 2n

(n ∈ N0 ),

(57)

0

where the coefficients an are defined by   ∞ 2−2z 0 12 − z X := an zn √ π 0(1 − z) n=0

and given by the following simple recursion formula: and (n + 1) an+1 = 2

a1 = 0

n X

  an−k 2k − 1 ζ (k + 1)

(n ∈ N).

k=1

From the equation (7.160) in [752, p. 230], the following formulas can be obtained: π

Z3   π 2m+1 x m (−1) x− log 2 sin dx 3 2 0

   1 = − (2m + 1)! 1 − 2−2m−2 1 − 3−2m−2 ζ (2m + 3) 2 m  π 2k ζ (2m + 3 − 2k) X + (2m + 1)! (−1)k (m ∈ N0 ); 3 (2k)!

(58)

k=0

π

Z3  log 0

sin x sin (x + π/3)

+ (2m + 2)!

m X k=0

2m+1

dx = (−1)m+1



 3m + 4  π 2m+3 2m + 3 3

 π 2k+1 ζ (2m + 2 − 2k) (−1)k 3 (2k + 1)!

(m ∈ N0 );

(59)

Miscellaneous Results π

(−1)m+1

Z3  0

= (2m)!

581

π !#2m Z3 "  π 2m x 1 sin x
log x− log 2 sin dx + dx 3 2 2m + 1 sin x + π3

0

m X k=1

 π 2k−1 ζ (2m + 3 − 2k) (−1)k 3 (2k − 1)!

(m ∈ N);

(60)

and m X

   π 2k ζ (2m + 2 − 2k) 1 6m + 5  π 2m+2 = (−1)m 3 (2k)! 2 (2m + 2)! 3 k=0     1 + 1 − 2−2m−1 1 − 3−2m−1 ζ (2m + 2) (m ∈ N). 2 (−1)k

(61)

The special cases of (58) and (59) when m = 0 yield, respectively, π

Z3   π x 2 x− log 2 sin dx = ζ (3) 3 2 3

(62)

0

and π

Z3 0

sin x
log sin (x + π3

! dx =

5π3 , 81

(63)

which is a corrected version of the second equation in [752, p. 230, Eq. (7.161)]. The formula (62) is recorded in [752, p. 230] and can be obtained from (51) and (52).

7.3 Applications of the Gamma and Polygamma Functions Involving Convolutions of the Rayleigh Functions The Gamma function and its nth logarithmic derivatives (i.e., the Polygamma or the Psi functions), which were introduced in Chapter 1, have found many interesting and useful applications in a variety of subjects in pure and applied mathematics. Choi and Srivastava [299] mainly applied these functions to treat convolutions of the Rayleigh functions by recalling a general identity expressing a certain class of series as Psifunctions and to evaluate a class of log-sine integrals in an algorithmic way. They [299] also evaluated some Euler sums and gave much simpler Psi-function expressions for some known parameterized multiple sums. Here, we choose to discuss only the application to convolutions of the Rayleigh functions in [299].

582

Zeta and q-Zeta Functions and Associated Series and Integrals

The Gamma function with its several equivalent forms (see Section 1.1) and the polygamma functions given in 1.3(52) have been used, in the vast mathematical literature, in many different ways. For example, the generalized Goldbach-Euler series has been shown to be connected with the ψ-function as follows (cf. [821, p. 88, Eq. (5)]): ∞ X ∞ X n=2 k=0

     1 r r−1 1 ψ −ψ , = (pk + r)n − 1 p p p

(1)

where p ∈ N and r = p or r = p + 1; we exclude the case r = p = 1. For other applications, see the works by (for example) Doelder [391] and Shen [1024], as well as others.

Series Expressible in Terms of the ψ-Function We recall one of the main results of Al-Saqabi et al. [21] as Theorem 7.1 below. Theorem 7.1. (Al-Saqabi et al. [21]) Suppose that P(x) is a polynomial in x of degree 5 r − 2 (r ∈ N \ {1}). Then, ∞ X

P(n) (λ1 + µ1 n) · · · (λr + µr n) n=1    Y  r r X λk λk ψ 1 + =− (λj µk − λk µj )−1 , µr−2 P − k µk µk

(2)

j=1 ( j6=k)

k=1

provided that the parameters λ1 , . . . , λr and µ1 , . . . , µr are constrained by λj µk 6= λk µj

( j 6= k;

j, k = 1, . . . , r).

An interesting generalization of Theorem 7.1 was derived by Wu et al. [1237]. Here, in our present investigation, we write a special case of (2) when P(x) ≡ 1 as follows: ∞ X k=0

 −1   r r Y X λj 1   =− µj uj ψ , (µ1 k + λ1 ) (µ2 k + λ2 ) · · · (µr k + λr ) µj j=1

j=1

(3) where r ∈ N \ {1}, µj 6= 0 ( j = 1, 2, . . . , r), λj /µj ’s are distinct complex numbers and 

−1

  Y λ λj  i   uj =  −   µi µj  15i5r i6=j

( j = 1, . . . , r).

Miscellaneous Results

583

The following further special case of (3) when r = 3 is recorded by Hansen [531, p. 106, Entry (6.1.65)]: ∞ X k=0

µ1 ψ(λ1 /µ1 ) 1 = (µ1 k + λ1 ) (µ2 k + λ2 ) (µ3 k + λ3 ) (µ2 λ1 − µ1 λ2 ) (µ1 λ3 − µ3 λ1 )

µ2 ψ(λ2 /µ2 ) µ3 ψ(λ3 /µ3 ) + + . (µ1 λ2 − µ2 λ1 ) (µ2 λ3 − µ3 λ2 ) (µ1 λ3 − µ3 λ1 ) (µ3 λ2 − µ2 λ3 )

(4)

Upon setting µj = 1 ( j = 1, 2, 3), λ1 = a, λ2 = b and λ3 = c, (4) yields ∞ X k=1

1 (k + a)(k + b)(k + c)

ψ(a) ψ(b) ψ(c) 1 = + + − . (a − b)(c − a) (b − a)(c − b) (c − a)(b − c) abc

(5)

Now, taking partial derivatives in (5) with respect to the variables a, b and c, we can express the following sum: ∞ X

1

k=1

(k + a)i (k + b)j (k + c)k

(i, j, k ∈ N)

(6)

in terms of the Psi-function and the polygamma functions. Only the case i = j = 3 and k = 1 of (6) is recorded here for our use in the next subsection: ∞ X

1

k=1

(k + a)3 (k + b)3 (k + c)

=

ψ 00 (a) ψ 00 (b) + 3 2 (a − b) (c − a) 2 (b − a)3 (c − b)

+

4a − b − 3c 4b − a − 3c ψ 0 (a) + ψ 0 (b) 4 2 (a − b) (a − c) (b − a)4 (b − c)2

+

6(c − a)2 − 3(a − b)(c − a) + (a − b)2 ψ(a) (a − b)5 (c − a)3

+

6(c − b)2 − 3(b − a)(c − b) + (b − a)2 ψ(b) (b − a)5 (c − b)3

+

ψ(c) 1 − 3 3 . 3 3 (c − a) (b − c) a b c

(7)

It may be remarked in passing that all of the interesting known or new special cases of the results, including (2) of Al-Saqabi et al. [21] (especially those that can be found in Hansen [531], Chrystal [313], Jolley [613] and Prudnikov et al. [918]), were mentioned explicitly by Al-Saqabi et al. [21]. Of course, Eq. (7) may be derived from a very specialized case of Eq. (25) in Wu et al. [1237, p. 6].

584

Zeta and q-Zeta Functions and Associated Series and Integrals

Convolutions of the Rayleigh Functions Varlamov [1192, 1193] systematically investigated convolutions of the Rayleigh functions with respect to the Bessel index and attempted to exhibit their usefulness for constructing global-in-time solutions of semilinear evolution equations in circular domains. Convolutions of the Rayleigh functions with respect to the Bessel index are defined as follows: Rl (m) =

∞ X

σl (m − k) σl (k)

(l ∈ N; m ∈ Z),

k=−∞

where Rayleigh functions σl (v) are defined as (see [1215, p. 502]) σl (v) =

∞ X 1 λ2l v,n

(l, n ∈ N),

n=1

v being the index of the Bessel function Jv (x), whose zeros are λv,n . Varlamov [1192, 1193] succeeded in treating the Rayleigh functions, σl (v) as special functions, by presenting some references for functions of the Rayleigh type, as well as in their own right (see the references cited by Varlamov [1192, 1193]). Varlamov [1192] expressed R1 (m) in terms of the ψ-function and made a simple use of the expression of the ψ-function to give the following asymptotic formula for R1 (m): R1 (m) ∼

ln |m| 4 |m|

(|m| → ∞).

Varlamov [1193] also presented a general formula for Rl (m) in terms of σl (v) and expressed R2 (m) in terms of the ψ-function, together with its asymptotic formula as |m| → ∞, by making use of the ψ-function, as follows:  2   2  1 2π 1 2π 4 R2 (m) ∼ 8 −5 − −5 3 3 3 2 |m| |m|4   2 34π /3 − 165 + 24(γ + ln |m|) ln |m| + + O (|m| → ∞). |m|5 |m|6 Furthermore, Varlamov [1193] provided an example of an initial-boundary-value problem leading to an explicit appearance of R1 (m) and R2 (m). To show a usefulness of the polygamma functions, we express R3 (m) in terms of the polygamma functions and give its asymptotic formula as |m| → ∞. It is easy to get the following formula of R3 (m) from one of Varlamov’s results [1193, Eq. (4.1)]: R3 (m) =

1 {A3 (m) + 2 B3 (m)} 210

(m ∈ Z),

(8)

Miscellaneous Results

585

where A3 (m) =

|m| X

1

k=0

(k + 1)3 (k + 2) (k + 3) (|m| − k + 1)3 (|m| − k + 2) (|m| − k + 3)

and B3 (m) =

∞ X

1

k=1

(k + 1)3 (k + 2) (k + 3) (|m| + k + 1)3 (|m| + k + 2) (|m| + k + 3)

.

We can, now, express A3 (m) and B3 (m) in terms of the polygamma functions. We, first, decompose A3 (m) into partial fractions as follows: A3 (m) =

n 1 (3|m|3 + 17|m|2 + 14|m| − 28)2 4(|m| + 2)5 (|m| + 3)3 (|m| + 4)3 |m|  o X − 2(|m| + 3) (|m| + 4) |m|4 − 4|m|3 − 62|m|2 − 150|m| − 88 k=0

−

1 k+1

|m| 3|m|3 + 17|m|2 + 14|m| − 28 X 1 4 2 2 2(|m| + 2) (|m| + 3) (|m| + 4) (k + 1)2 k=0

+

−

1 (|m| + 2)3 (|m| + 3) (|m| + 4)

|m| X k=0

1 (k + 1)3

|m| X 2 1 k+2 (|m| + 3)3 (|m| + 4) (|m| + 5) k=0

+

|m| X 1 1 , 3 k+3 4(|m| + 4) (|m| + 5) (|m| + 6) k=0

so that, by using 1.2(7), we have n 1 (3|m|3 + 17|m|2 + 14|m| − 28)2 4(|m| + 2)5 (|m| + 3)3 (|m| + 4)3  o −2(|m| + 3) (|m| + 4) |m|4 − 4|m|3 − 62|m|2 − 150|m| − 88 (ψ(|m| + 2) + γ )

A3 (m) =

3|m|3 + 17|m|2 + 14|m| − 28 (ψ 0 (|m| + 2) − ψ 0 (1)) 2(|m| + 2)4 (|m| + 3)2 (|m| + 4)2 1 + (ψ 00 (|m| + 2) − ψ 00 (1)) 2(|m| + 2)3 (|m| + 3) (|m| + 4) −

586

Zeta and q-Zeta Functions and Associated Series and Integrals

− +

2 (|m| + 3)3 (|m| + 4) (|m| + 5)

(ψ(|m| + 3) − ψ(2))

1 (ψ(|m| + 4) − ψ(3)), 4(|m| + 4)3 (|m| + 5) (|m| + 6)

where γ denotes the Euler-Mascheroni constant, defined by 1.1(3). Next, we decompose B3 (m) into the following partial fractions: B3 (m) =

S1 (m) S2 (m) S3 (m) S4 (m) − + − , |m|(|m| + 1) |m|(|m| − 1) |m|(|m| − 1) |m|(|m| + 1)

where S1 (m) = S2 (m) = S3 (m) =

∞ X k=1 ∞ X k=1 ∞ X k=1

1 , (k + 1)3 (k + |m| + 1)3 (k + 2) 1 (k + 1)3 (k + |m| + 1)3 (k + 3)

,

1 , (k + 1)3 (k + |m| + 1)3 (k + |m| + 2)

and S4 (m) =

∞ X k=1

1 . (k + 1)3 (k + |m| + 1)3 (k + |m| + 3)

Using (7), 1.3(53) and other needed identities listed in Chapters 1 and 2, we can express Si (m) (i = 1, 2, 3, 4) as follows: 4|m| − 3 ζ (3, |m| + 1) π 2 |m| + 3 − + ζ (2, |m| + 1) 6 |m|4 |m|3 (|m| − 1) |m|4 (|m| − 1)2 γ (|m|2 + 3|m| + 6) 10 |m|2 − 15 |m| + 6 ψ(|m| + 1) + − |m|5 |m|5 (|m| − 1)3 ζ (3) 1−γ 1 + + − . 3 3 |m| (|m| − 1) 2 (|m| + 1)3 ζ (3, |m| + 1) π 2 |m| + 6 2(2|m| − 3) S2 (m) = − + ζ (2, |m| + 1) |m|3 (|m| − 2) 24 |m|4 |m|4 (|m| − 2)2 γ |m|2 + 6|m| + 24 2(2|m|2 − 15|m| + 12) + − ψ(|m| + 1) 8 |m|5 |m|5 (|m| − 2)3 ζ (3) 3 − 2γ 1 + + − . 2 |m|3 16(|m| − 2)3 3 (|m| + 1)3 S1 (m) =

Miscellaneous Results

S3 (m) =

587

ζ (3) ζ (3, |m| + 1) π 2 4|m| + 3 − − 3 3 6 |m|4 (|m| + 1)2 |m| (|m| + 1) |m| |m| − 3 γ (10 |m|2 + 15 |m| + 6) + ζ (2, |m| + 1) + |m|4 |m|5 (|m| + 1)3 2 |m| − 3|m| + 6 ψ(|m| + 2) 1 − . + ψ(|m| + 1) − 5 3 3 |m| (|m| + 1) (|m| + 1) (|m| + 2) |m| − 6 ζ (3, |m| + 1) π 2 2|m| + 3 − + ζ (2, |m| + 1) 3 4 2 3 |m| (|m| + 2) 2 |m| 4 |m|4

S4 (m) = −

2γ (5 |m|2 + 15 |m| + 12) |m|2 − 6|m| + 24 + ψ(|m| + 1) 8 |m|5 |m|5 (|m| + 2)3 ζ (3) ψ(|m| + 3) 1 + − − . |m|3 (|m| + 2) 8 (|m| + 2)3 (|m| + 1)3 (|m| + 3) +

Finally, by applying 1.3(65) to each of the above expressions, we get the following asymptotic formulas: A3 (m) =

   13 π 2 1 1 + + ζ (3) +O 8 4 |m|5 |m|6

(|m| → ∞)

31 π 2 ζ (3) − + 48 8 2

(|m| → ∞),



and B3 (m) =





  1 1 +O |m|5 |m|6

which, upon using (8), yields a simple asymptotic formula for R3 (m) as follows: 1 R3 (m) = 10 2



35 + 2 ζ (3) 12



  1 1 +O |m|5 |m|6

(|m| → ∞).

7.4 Bernoulli and Euler Polynomials at Rational Arguments We begin by recalling a theorem involving the Bernoulli numbers discovered in 1840 (and proven independently and published almost simultaneously) by C. von Staudt (1798–1867) and T. Clausen (1801–1885). Theorem 7.2. [von Staudt-Clausen Theorem] Let Bn denote the Bernoulli numbers, defined by 1.7(2). Then, B2n = In −

X 1 p

(n ∈ N),

(1)

p−1|2n

where In is an integer and the sum is taken over all primes p, such that p − 1 divides 2n.

588

Zeta and q-Zeta Functions and Associated Series and Integrals

As noted by Carathe´ odory [210, p. 283], the von Staudt-Clausen theorem (1) is very useful for the calculation of higher-order Bernoulli numbers, since the value of In follows even from a preliminary estimate of B2n that one might obtain in one way or another (see also Problem 6 of Chapter 1). Bartz and Rutkowski [99] gave a simple proof of a generalization of the von StaudtClausen theorem (1) to hold true for the Bernoulli polynomials, which also reduces to another interesting and inspiring result (Almkvist and Meurman [19, p. 104, Theorem 2]): i h   kn Bn hk − Bn ∈ Z (h, k ∈ N),

(2)

by Almkvist and Meurman [19, p. 104, Theorem 2]), who proved (2) in an elementary (yet rather complicated) way. Subsequently, simpler proofs of the Almkvist-Meurman theorem (2) have been given by Sury [1137] and Clarke and Slavutsk II [1041]. An analogous result involving the Euler polynomials, instead, was proven by Fox [459]. All the aforecited works are concerned with evaluations of the Bernoulli and Euler polynomials at rational arguments. The remainder of this section is based largely upon the work of Srivastava [1087], who gave several remarkably shorter proofs of each of the Cvijovic´ -Klinowski summation formulas and also considered their various generalizations and analogues.

The Cvijovi´c-Klinowski Summation Formulas By employing a number of properties and characteristics of certain Dirichlet and trigonometric series, Cvijovic´ and Klinowski [349] evaluated the Bernoulli polynomials Bn (x) with n ∈ N \ {1} and the Euler polynomials En (x) with n ∈ N for 0 5 x 5 1 when x is a rational number. For the sake of ready reference, we recall, here, the main results of Cvijovic´ and Klinowski [349] as Theorem 7.3 and Theorem 7.4 below: Theorem 7.3 In terms of the Hurwitz Zeta function ζ (s, a), defined by 2.2(1), the Bernoulli polynomials Bn (x) at rational arguments are given by B2n−1

      q 2(2n − 1)! X j 2jpπ p = (−1)n ζ 2n − 1, sin q q q (2qπ)2n−1 j=1

(3)

(n ∈ N \ {1}; p ∈ N0 ; q ∈ N; 0 ≤ p ≤ q) and B2n

      q p j 2jpπ 2(2n)! X = (−1)n−1 ζ 2n, cos q q q (2qπ)2n j=1


n ∈ N; p ∈ N0 ; q ∈ N; 0 5 p 5 q .

(4)

Miscellaneous Results

589

Theorem 7.4 In terms of the Hurwitz Zeta function ζ (s, a), defined by 2.2(1), the Euler polynomials En (x) at rational arguments are given by E2n−1

      q 2j − 1 4(2n − 1)! X (2j − 1)pπ p ζ 2n, = (−1)n cos q q q (2qπ)2n

(5)

j=1




n ∈ N; p ∈ N0 ; q ∈ N; 0 5 p 5 q and E2n

      q 4(2n)! X 2j − 1 (2j − 1)pπ p = (−1)n ζ 2n + 1, sin q 2q q (2qπ)2n+1 j=1
n ∈ N; p ∈ N0 ; q ∈ N; 0 5 p 5 q .

(6)

As pointed out by Cvijovic´ and Klinowski [349, p. 1535], the formula (3) was derived earlier, in a completely different way, by Almkvist and Meurman [19, p. 107, Proposition 10], who applied this formula to show eventually that their assertion holds true.

Srivastava’s Shorter Proofs of Theorem 7.3 and Theorem 7.4 First Proof of Theorem 7.3. We begin by recalling the following known result (cf., e.g., Magnus et al. [795, p. 27]): ∞

Bn (x) = −

 2 · n! X 1 nπ  cos 2kπ x − (2π)n kn 2

(7)

k=1

n ∈ N \ {1} and

0 5 x 5 1; n = 1

and

0<x<1 ,


which, in view of the definition in 2.2(1) and the elementary series identity: ∞ X

f (k) =

q X ∞ X

f (qk + j)

(q ∈ N),

(8)

j=1 k=0

k=1

immediately yields       q p 2 · n! X j 2jpπ nπ ζ n, − Bn =− cos q (2qπ)n q q 2

(9)

j=1

n ∈ N \ {1}; p ∈ N0 ; q ∈ N; 0 5 p 5 q ,


where we have also set x=

p q


p ∈ N0 ; q ∈ N; 0 5 p 5 q .

590

Zeta and q-Zeta Functions and Associated Series and Integrals

Upon replacing n in (9) by 2n − 1 and 2n, respectively, we obtain the assertions (3) and (4) of Theorem 7.3.  Second Proof of Theorem 7.3. Upon setting s = n (n ∈ N \ {1}) in 2.2(8) and applying 2.1(17) (with ` = n − 1) to the left member of the resulting identity, we arrive once again at the formula (9), which unifies the assertions (3) and (4) of Theorem 7.3, the missing case p = 0 for the Bernoulli numbers Bn being easily verified directly by means of the well-known identities 2.3(18) and ζ (s) = q

−s

  q X j ζ s, q

(q ∈ N).

(10)

j=1



First Proof of Theorem 7.4. We, now, show that Theorem 7.4 is actually a simple consequence of Theorem 7.3. Indeed, by appealing to the known relationship 1.7(60), we find from (9) (with n replaced by n + 1) that        q 2  2(n + 1)! X 2jpπ nπ p j = sin − − En ζ n + 1, q n+1 q q 2 (2qπ)n+1 j=1     2q  X 2(n + 1)! j jpπ nπ  + 2n+1 ζ n + 1, sin − 2q q 2 (4qπ)n+1 j=1     2q  4 · n! X j jpπ nπ sin = ζ n + 1, − 2q q 2 (2qπ)n+1 j=1      q X j 2jpπ nπ  − ζ n + 1, sin − p q 2

(11)

j=1


n ∈ N; p ∈ N0 ; q ∈ N; 0 5 p 5 q . Obviously, since    2q  X j jpπ nπ ζ n + 1, sin − 2q q 2 j=1

    q X 2j − 1 (2j − 1)pπ nπ = ζ n + 1, sin − 2q q 2 j=1

    q X j 2jpπ nπ + ζ n + 1, sin − , q q 2 j=1

(12)

Miscellaneous Results

591

upon separating the even and odd terms, (11) leads us immediately to the formula: En

      q 2j − 1 4 · n! X (2j − 1)pπ nπ p ζ n + 1, = sin − q 2q q 2 (2qπ)n+1 j=1

(13)


n ∈ N; p ∈ N0 ; q ∈ N; 0 5 p 5 q , which yields the assertions (5) and (6) of Theorem 7.3 when n is replaced by 2n − 1 and 2n, respectively.  Second Proof of Theorem 7.4. Just as in our first proof of Theorem 7.3, the unification (13) of the assertions (5) and (6) of Theorem 7.4 can be proven directly by merely applying the series identity (8) in the following known result (cf., e.g., Magnus et al. [795, p. 30]; see also 1.7(56) and 1.7(57)): ∞

En (x) =

 1 nπ  4 · n! X sin (2k − 1)πx − 2 π n+1 (2k − 1)n+1

(14)

k=1

(n ∈ N and

0 5 x 5 1; n = 0

and

0 < x < 1),

with, of course, x=

p q


p ∈ N; q ∈ N; 0 5 p 5 q .



Formulas Involving the Hurwitz-Lerch Zeta Function By virtue of 1.3(53), each of the formulas (9) and (13) and their even and odd integer versions given by Theorem 7.3 and Theorem 7.4 can easily be restated in terms of the Polygamma function. For the general Hurwitz-Lerch Zeta function 8(z, s, a), defined by 2.5(1), it is easily seen by using the series identity (8) that 8(z, s, a) = q−s

q X j=1

  a + j − 1 j−1 z , 8 zq , s, q

(15)

which, in the special case when  z = exp

2pπi q



(p ∈ Z; q ∈ N),

yields the summation formula:      q X p a+j−1 2( j − 1)pπi −s L , s, a = q ζ s, exp q q q 

j=1

(16)

592

Zeta and q-Zeta Functions and Associated Series and Integrals

for the generalized Zeta function L(x, s, a), defined by 2.5(11), in terms of the Hurwitz Zeta function ζ (s, a). For z = 1, (15) reduces at once to the identity [Hansen (1975, p. 360, Entry (54.13.1))]: ζ (s, a) = q

−s

  q X a+j−1 , ζ s, q

(17)

j=1

which, for a = 1, yields the well-known result (10). Conversely, by setting a = (15) and (16), we have ∞ X n=1

  q X 2j − 1 j−1 zn −s q = (2q) 8 z , s, z (2n − 1)s 2q

1 2

in

(18)

j=1

and ∞ (2n+1)pπ i/q X e n=0

(2n + 1)s

= (2q)−s

    q X 2j − 1 (2j − 1)pπi ζ s, exp , 2q q

(19)

j=1

respectively. Lastly, in their special cases when a = 1, (15) and (16) yield the following companions of the summation formulas (18) and (19), respectively: ∞ n X z n=1

ns

=: Lis (z) = q

−s

q X j=1

 j j 8 z , s, z q 

q

(20)

and ∞ 2npπ i/q X e n=1

ns

     q X p j 2jpπi −s =: L ,s = q ζ s, exp . q q q 

(21)

j=1

In particular, when s = ν (ν > 1), by simply equating the real and imaginary parts in (19) and (21), we immediately obtain the following summation formulas involving trigonometric series: ∞ X cos [(2n + 1)pπ/q] n=0

(2n + 1)ν

∞ X sin [(2n + 1)pπ/q] n=0

(2n + 1)ν

= (2q)

−ν

= (2q)

−ν

    q X 2j − 1 (2j − 1)pπ ζ ν, cos ; 2q q

j=1 q X j=1

   2j − 1 (2j − 1)pπ ζ ν, sin ; 2q q

(22)



(23)

Miscellaneous Results

593

∞ X cos(2npπ/q)

nν

n=1

∞ X sin(2npπ/q)

nν

n=1

= q−ν = q−ν

    q X j 2jpπ ζ ν, cos ; q q

j=1 q X j=1

    j 2jpπ ζ ν, sin . q q

(24)

(25)

The assertions made by the Lemma of Cvijovic´ and Klinowski [349, p. 1530] are precisely the special summation formulas (19) and (21) with s = ν (ν > 1). Formula (21) with s = ν (ν > 1) is, in fact, also one of the three main results in another paper by Cvijovic´ and Klinowski [350, p. 207, Eq. (7)]; the other two main results of Cvijovic´ and Klinowski [350, p. 207, Eqs. (8a) and (8b)] are essentially the same as (23) and (22), respectively, with p replaced by 2p. Cvijovic´ and Klinowski [350, p. 208, Eqs. (9a) and (9b)] also gave the special summation formulas (23) and (22), respectively, with q = Q. As already pointed out by Srivastava [1087], each of the trigonometric sums (22) to (25), and indeed also various further special cases of many of the summation formulas considered here, can be found derived in the work of Cvijovic´ and Klinowski [349].

An Application of Lerch’s Functional Equation 2.5(29) In view of the summation formula (16), Lerch’s functional equation 2.5(29) with a=

p q

(p ∈ Z; q ∈ N)

yields       q p 0(s) X x+j−1 s 2(x + j − 1)p L x, 1 − s, − = ζ s, exp πi q (2pπ)s  q 2 q j=1       q X j−x s 2( j − x)p + ζ s, exp − + πi . (26)  q 2 q 



j=1

In terms of a mild generalization Bn (a, α) of the classical Bernoulli polynomials Bn (x), defined by 2.5(41), by setting s = n (n ∈ N) in (26) and using 2.5(40), Srivastava [1087] obtained the following novel analogue of (9) for the generalized Bernoulli polynomials Bn (a, α):         q p 2πix x+j−1 n 2(x + j − 1)p n! X Bn ,e ζ n, exp − πi =− q (2qπ)n  q 2 q j=1       q X j−x n 2( j − x)p + ζ n, exp − + πi , (27)  q 2 q j=1

594

Zeta and q-Zeta Functions and Associated Series and Integrals

which holds true whenever each side exists. Indeed, in its special case when x ∈ Z, Srivastava’s formula (27) can easily be shown to reduce to the known result (9).

7.5 Closed-Form Summation of Trigonometric Series By Weierstrass’s M-test, it is easily seen that each of the classical trigonometric series:

Cν (α) :=

∞ X cos(2k + 1)α (2k + 1)ν

and Sν (α) :=

k=0

∞ X sin(2k + 1)α (2k + 1)ν

(1)

k=0

converges uniformly for all real values of α when ν > 1. Furthermore, since Cν (−α) = Cν (α) = −Cν (π − α)

(2)

Sν (−α) = −Sν (α) = −Sν (π − α),

(3)

and

it is sufficient to evaluate Cν (α) and Sν (α) only over the range 0 5 α 5 12 π. The series C2n+1 (α) and S2n (α) (n ∈ N) converge extremely slowly, and the problem of their numerical evaluation was addressed by (among others) Dempsey et al. [373], Boersma and Dempsey [139] and Gautschi [473]. For the particular case when n = 1, Dempsey et al. [373] developed a procedure based on Plana’s summation formula, together with Romberg’s method of integration, which significantly improves the convergence and accuracy when compared with those resulting from direct summation. Boersma and Dempsey [139], conversely, transformed each of C2n+1 (α) and S2n (α) into a rapidly convergent series that is well-suited for their computation (when, e.g., n = 1). Recently, Cvijovic´ and Klinowski [350] considered the general series in (1) when ν > 1 and α is a rational multiple of 2π. In terms of the Riemann Zeta function ζ (s) and the Hurwitz Zeta function ζ (s, a), defined by 2.3(1) and 2.2(1), respectively, they first showed that (Cvijovic´ and Klinowski [350, p. 207, Eqs. (8a) and (8b)])      2pπ 1 1 = ν ζ (ν) 1 − ν Cν q q 2        (4) q−1 X j 2jpπ 1 4jpπ + ζ ν, cos − ν cos q q 2 q j=1

and  Sν

2pπ q



     q−1  1 X j 2jpπ 1 4jpπ = ν ζ ν, sin − ν sin q q q 2 q j=1

(p, q ∈ N; ν > 1).

(5)

Miscellaneous Results

595

Following the work of Srivastava [1085]), we now show that the summation formulas (4) and (5), as well as their equivalent variants when α is a rational multiple of π (cf. Cvijovic´ and Klinowski [350, p. 208, Eqs. (9a) and (9b)]), would follow easily from some general results involving the family of Dirichlet series in 2.5(1), which is usually called Lerch’s function or Lerch’s transcendent. For convenience, we recall, here, the summation formulas 6.1(24), 6.1(25), 6.1(22) and 6.1(23) in the following (slightly modified) form:     q ∞ X X j 2jpπ cos(2kpπ/q) −ν = (2q) ζ ν, cos , (2k)ν q q

(6)

∞ X sin(2kpπ/q) = (2q)−ν (2k)ν

(7)

k=1

k=1

 Cν

pπ q



j=1 q X j=1

    j 2jpπ ζ ν, sin , q q

∞ X cos [(2k + 1)pπ/q] (2k + 1)ν k=0     q X 2j − 1 (2j − 1)pπ −ν = (2q) ζ ν, cos , 2q q

:=

(8)

j=1

and  Sν

pπ q

∞ X sin [(2k + 1)pπ/q] (2k + 1)ν k=0     q X 2j − 1 (2j − 1)pπ ζ ν, sin , = (2q)−ν 2q q



:=

(9)

j=1

respectively. Since (cf. Equation 6.1(17) with q = 2) ζ (s, a) = 2

−s

     1 1 1 ζ s, a + ζ s, a + , 2 2 2

the second member of the summation formula (4) can be rewritten in the form: q−ν

      q X 2jpπ 4jpπ j ζ ν, cos − 2−ν cos q q q j=1

= (2q)

−ν

X      q   j+q 2jpπ j + ζ ν, cos ζ ν, 2q 2q q j=1

    q X j 4jpπ − ζ ν, cos q q j=1

(10)

596

Zeta and q-Zeta Functions and Associated Series and Integrals

= (2q)−ν

 X    q 2j − 1 2(2j − 1)pπ ζ ν, cos 2q q

(11)

j=1

+

−

    q X 2j 4jpπ ζ ν, cos 2q q

j=1 q X j=1

    4jpπ j cos ζ ν, q q

    q X 2j − 1 2(2j − 1)pπ −ν = (2q) ζ ν, cos , 2q q j=1

where we have also applied the trigonometric identity:  cos

2jpπ q



 = cos

2( j + q)pπ q




1 5 j 5 q; j, q ∈ N; p ∈ Z .

Thus, the summation formula (4) is reduced to its equivalent form:  Cν

2pπ q



= (2q)−ν

    q X 2j − 1 2(2j − 1)pπ ζ ν, cos , 2q q

(12)

j=1

which is precisely (8) with p replaced rather trivially by 2p. Next, since the j = q term in (5) is obviously nil, it is not difficult to reduce the summation formula (5) similarly to its equivalent form:  Sν

2pπ q



= (2q)

−ν

    q X 2j − 1 2(2j − 1)pπ ζ ν, sin , 2q q

(13)

j=1

which is precisely (9) with p replaced rather trivially by 2p. Each of the summation formulas (6) to (9) can, indeed, be proven directly by means of the series identity 6.1(8). The derivation of (4) and (5), by Cvijovic´ and Klinowski [350], made use of the relationship: χν (z) = Liν (z) − 2−ν Liν (z2 ),

(14)

where χν (z) :=

∞ X k=0

z2k+1 (2k + 1)ν

and

Liν (z) :=

∞ k X z kν k=1

|z| 5 1; <(ν) > 1




(15)

Miscellaneous Results

597

denote Legendre’s Chi function and the Polylogarithm function of order ν (cf., e.g., Lewin [752]). In fact, as observed by Cvijovic´ and Klinowski [350, p. 208], the (equivalent) summation formulas (8) and (9) are obtained when we make use of the relationship: 1 [Liν (z) − Liν (−z)] , 2

χν (z) =

(16)

instead of (14).

Problems 1. Borwein et al. [145] presented some very interesting parameterized classes of multiple sums whose many special cases reduce to known Euler and related sums, by making a considerable use of computer algebra systems (as they noted). Their main formulas are recalled here as follows: ∞ X n=1 ∞ X n=1

n−1 ∞ X X 1 1 1 = n(n − x) m−x n2 (n − x) m=1

(x ∈ C \ N);

(a)

n=1

n−1 X X 1 my = ζ (2s, 2t + 1) x2s−2 y2t+1 n2 − x2 m 2 − y2 m=1

s>0, t=0

=

ζ (s, t) :=

X n>m=1

π [x cot(π x) + 2 y cot(π y)] S1 (x, y) 2 π 1 + x cot(π x) S2 (x, y) − S3 (x, y) 2 2
x, y ∈ C \ (Z \ {0}) ;

(b)

∞ n−1 X 1 X 1 1 = ; ns mt ns mt n=1

S1 (x, y) := S2 (x, y) :=

 n=1 ∞ X n=1

S3 (x, y) :=

m=1

∞ X

∞ X n=1

ny 

; (n − x)2 − y2
ny 4y2 − x2
  ; n2 − y2 (n + x)2 − y2 (n − x)2 − y2

(n + x)2 − y2

n 2 − y2

ny


n2 − x2


.

In fact, Borwein et al. [145] expressed Sj (x, y) ( j = 1, 2, 3) in terms of the ψ-function and cot(x). Show that Sj (x, y) ( j = 1, 2, 3) can be expressed in terms of the psi-function, only in a much simpler way as follows: S1 (x, y) = −

1 [ψ(1 + x + y) − ψ(1 + x − y) − ψ(1 − x + y) + ψ(1 − x − y)] ; 8x

598

Zeta and q-Zeta Functions and Associated Series and Integrals

x + 2y [ψ(1 + x − y) − ψ(1 − x + y) + ψ(1 − y) − ψ(1 + y)] 8 x2 x − 2y [ψ(1 − x − y) − ψ(1 + x + y) + ψ(1 − y) − ψ(1 + y)] ; + 8 x2 y
[ψ(1 + x) + ψ(1 − x) − ψ(1 + y) − ψ(1 − y)] . S3 (x, y) = 2 y2 − x 2

S2 (x, y) =

(Choi and Srivastava [299]) 2. Upon replacing x by t in (a) and integrating the resulting identity with respect to t from 0 to x, Borwein et al. [145] found that   ∞ ∞ n−1  X 1 x X 1 X 1 1 − x/m log 1 − = log n n n−m 1 − x/n n2 n=1

n=1

(x ∈ C \ N).

(c)

m=1

Use (c) and prove the following formulas:   X ∞ n−1 ∞ X 1 X 1 m arctan(1/n) arctan 2 = n m n − mn + 1 n2 n=1

m=1

n=1

∞ X (−1)k = ζ (2k + 3) 2k + 1 k=0

and  ∞ ∞ n−1 X X n2 (n − m)2 + 1 1 X 1 (−1)k−1 = log ζ (2k + 2). 2 2 n m k (n − m) (n + 1) n=1

m=1

k=1

(Choi and Srivastava [299]; see also Borwein et al. [145, p. 331]) 3. Show that Z∞ 0

  1 1 P(x) dx = z − log z − z + log(2π ) − log 0(z), z+x 2 2

where P(x) is given as in (1) and |arg(z)| 5 π −  (0 <  < π). (Rainville [959, p. 30, Theorem 12]) 4. Show that the constant A4 in 7.1(42) with p = 4 can be expressed in the form: log A4 =

5 1 1 G + log 2 − log A − , 64 2 8 2π

where A is the Glaisher-Kinkelin constant, defined by 1.3(2), and G is the Catalan constant, defined by 1.3(16). (Choi and Srivastava [289, p. 100, Eq. (2.30)]) 5. Evaluate the following integral: Z∞ 2x − 1 P(x) log dx, 2x + 1 1

where P(x) = x − [x] − 21 . (Choi et al. [310, pp. 108–109, Eq. (3.7)]; Zhang [1249, p. 69, Eq. (14)])

Miscellaneous Results

599

6. Assume that n ∈ N and set w = exp(iπ/n). Let ζ (s, a) and χν (z) be the Hurwitz Zeta and Legendre Chi functions, defined by 2.2(1) and 6.2(15), respectively. Then show that   n   2j − 1 1X ζ ν, = (2n)ν χν wk w−k(2j−1) 2n n

( j = 1, . . . , n),

k=1

and   χν wk =

  n 2j − 1 1 X ζ ν, wk(2j−1) (2n)ν 2n

( j = 1, . . . , n).

j=1

(Cvijovi´c and Klinowski [352, p. 1625, Eqs. (7a) and (7b)]) 7. Prove that     2p − 1 2p − 1 + ζ 2n, 1 − ζ 2n, 2q 2q     q r (2q)2n X (2p − 1)rπ E2n−1 = (−1)n π 2n cos , (2n − 1)! q q r=1

and     2p − 1 2p − 1 ζ 2n + 1, − ζ 2n + 1, 1 − 2q 2q     q 2n X r (2p − 1)rπ n 2n+1 (2q) = (−1) π E2n sin (2n)! q q

(n ∈ N).

r=1

(Cvijovi´c and Klinowski [352, p. 1628, Eqs. (15a) and (15b)]) 8. Prove that, if f (x) ∈ Q[x] is integer-valued on N0 , that is, if f (k) ∈ Z for all k ∈ N0 , and p is prime, then f (a) ∈ Z(p) for all a ∈ Z(p) , where Z(p) denotes the ring of rational numbers whose denominator is not divisible by the prime p. (Clarke and Slavutsk II [325, p. 22, Proposition 1]) 9. Show that, for α > −1, Z1 0

q−1

1 X ρα dρ = 2q ρ 2 − 2ρ cos θ + 1 k=1



   sin(kθ ) α−k ψ 1+ , sin θ 2q

where θ=

pπ q

(p, q ∈ N; 0 < p < 2q; (p, q) = 1),

and (p, q) denotes that p and q are relatively prime. (de Doelder [391, p. 327])

600

Zeta and q-Zeta Functions and Associated Series and Integrals

10. Prove that  G 1+ log  G 1−

p 2q p 2q

  =−

+

   p 1 pπ log sin 2q π 2q       q−1  r 1 X r prπ 0 0 ψ 1 − − ψ sin 2q 2q q 8πq2 r=1

(p ∈ N; q ∈ N \ {1}; 1 5 p 5 2q − 1; (p, q) = 1), where G denotes the Barnes G-function considered in Section 1.3 and (p, q) is the same notation as in Problem 8. (Choi and Srivastava [289, p. 95, Eq. (2.2)]) 11. Evaluate the following integral: Z∞ 1

3 1 1 P(x) dx = − + log 2 + log 3, 2x + 1 4 4 2

where P(x) = x − [x] − 12 . (Zhang [1249, p. 68, Eq. (11)]; see also [1094, Section 6.3]) 12. Evaluate the following integrals: π

Zp

  x2m dx = 2 (−1)m (2m)! 1 − 2−2m−1 ζ (2m + 1) sin x

0

−

(2m)! p

 2m X     p−1 m π (−1)k (2π )−2k X (2` + 1)π 2` + 1 cos ζ 2k + 1, p (2m − 2k)! p 2p `=0

k=1

  2m+1 X    p−1 m (2` + 1)π π (−1)k (2π )−2k X 2` + 1 sin ζ 2k, − 2 (2m)! p (2m − 2k + 1)! p 2p `=0 k=1  2m π −2 γ (p) (m ∈ N; p ∈ N \ {1}) p and π

Zp

x2m+1 dx sin x

0

=

(2m + 1)! 2 p2

 2m X     p−1 m π (−1)k (2π )−2k X (2` + 1)π 2` + 1 sin ζ 2k + 2, p (2m − 2k)! p 2p k=0

`=0

      p−1 m (2m + 1)! π 2m+1 X (−1)k (2π )−2k X (2` + 1)π 2` + 1 − cos ζ 2k + 1, p p (2m − 2k + 1)! p 2p `=0 k=1  2m+1 π −2 γ (p) (m ∈ N0 ; p ∈ N \ {1}), p

Miscellaneous Results

601

where γ (p) is given by γ (p) =

p−1 ∞ X X j=0 `=0

  (2` + 1)π 1 cos 2pj + 2` + 1 p

and (as usual) an empty sum is understood to be nil. (Cho et al. [257, Equations (20) and (22)]) 13. Show that ∞ X

∞

Hn

n=1

X xn 1 xn = [log(1 − x)]2 + n 2 n2

(|x| < 1).

n=1

(Alzer et al. [31, Eq. (3.47)]; see also [531, p. 63, Entry (5.13.15)]) 14. Show that   ∞ n X 1 1 X 1 k n ζ (s) = (−1) k (k + 1)s 1 − 21−s 2n+1 n=0

(s ∈ C \ {1}).

k=0

(Coffey [330, Eq. (11)]; see also [542])

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